UG Mathematics Booster Test 3 - Operations on Matrices -

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UG Mathematics Booster Test 3 - Operations on Matrices

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

UG Mathematics Booster Test 3 - Operations on Matrices

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QUESTION 1 OF 20

Which statement analytically evaluates the "same order rule" INCORRECTLY? (Incorrect Statement)

Answer Review

1 Which statement analytically evaluates the "same order rule" INCORRECTLY? (Incorrect Statement)

A Two matrices of orders and cannot be algebraically added.

B The sum matrix inherently retains the exact same domain structure as the original matrices.

C A row matrix can be added to a column matrix if they simply have the same number of elements.

D Matrix addition is defined as a binary operation strictly on the set of matrices of the identical order.

💡 Short Explanation

Matrix addition requires identical order Same number of elements is insufficient Row and column structures must also match

✅ Correct Answer Explanation

Matrices can be added only when they have exactly the same order. A row matrix \(1\times n\) and a column matrix \(n\times 1\) have different structures, even if they contain the same number of elements. Therefore, Option C is incorrect. Options A, B, and D correctly describe the same-order rule for matrix addition.

❓ Why Other Options Are Incorrect

Option A → Correct because matrices of different orders cannot be added.
Option B → Correct since the resulting sum matrix retains the same order as the original matrices.
Option D → Correct because matrix addition is defined only for matrices of identical order.

🎯 Strategy

Used: Elimination

Application:

Remove statements consistent with matrix addition rules and identify the structurally incorrect one.

Final Logic:

Equal element count alone does not guarantee equal order.

🧠 Memory Trick

"Same shape, then add."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Addition of Matrices

2 Match each addition to its resulting element:

List 1:	List 2:
1. \(3+4\)	a. \(-7\)
2. \(-5+(-2)\)	b. \(7\)
3. \(1+2\)	c. \(3\)
4. \(5+(-2)\)	d. \(3\)

A 1-b, 2-a, 3-c, 4-d

B 1-a, 2-b, 3-c, 4-d

C 1-d, 2-a, 3-b, 4-c

D 1-c, 2-d, 3-a, 4-b

💡 Short Explanation

Add integers carefully Positive and negative rules apply Match results systematically

✅ Correct Answer Explanation

Evaluating each expression: \(3+4=7-5+(-2)=-71+2=35+(-2)=3\) Thus the correct matching is: 1-b, 2-a, 3-c, 4-d. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrectly matches \(3+4\) with \(-7\).
Option C → Swaps positive and negative results incorrectly.
Option D → Several addition results are mismatched mathematically.

🎯 Strategy

Used: Substitution

Application:

Directly compute each arithmetic expression and match values.

Final Logic:

Correct arithmetic gives the required matching pattern.

🧠 Memory Trick

"Positive adds, negatives reduce."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Element-wise Matrix Addition

3 Let
\(A=\left[\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}\right],k=\int_{0}^{1}\,2x dx=1\)
Find \(kA\).

A \(\left[\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}\right]\)

B \(\left[\begin{pmatrix}2 & 4\\ 6 & 8\end{pmatrix}\right]\)

C \(\left[\begin{pmatrix}0 & 0\\ 0 & 0\end{pmatrix}\right]\)

D \(\left[\begin{pmatrix}1 & 1\\ 1 & 1\end{pmatrix}\right]\)

💡 Short Explanation

Integral evaluates to 1 Multiplying by 1 leaves matrix unchanged Scalar identity property applies

✅ Correct Answer Explanation

Evaluate the integral: \(\int_{0}^{1}\,2x dx=[x^{2}]_{0}^{1}=1\) Thus \(k=1\). Scalar multiplication by 1 leaves every matrix unchanged: \(1A=A\) Therefore, Option A is correct. Other options represent multiplication by different scalars or unrelated matrices.

❓ Why Other Options Are Incorrect

Option B → Represents multiplication by scalar 2, not 1.
Option C → Zero matrix would occur only if scalar were 0.
Option D → Scalar multiplication does not replace all entries with 1.

🎯 Strategy

Used: Substitution

Application:

First evaluate the integral, then apply scalar multiplication rule.

Final Logic:

Since \(k=1\), matrix remains unchanged.

🧠 Memory Trick

"One keeps same."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Scalar Multiplication

4 Let
\(P=\left[\begin{pmatrix}0.2 & 0.3\end{pmatrix}\right],k=5\)
Find expected failures matrix \(E=kP\).

A \(\left[\begin{pmatrix}1.0 & 1.5\end{pmatrix}\right]\)

B \(\left[\begin{pmatrix}1 & 1.5\end{pmatrix}\right]\)

C \(\left[\begin{pmatrix}0.2 & 0.3\end{pmatrix}\right]\)

D \(\left[\begin{pmatrix}5 & 5\end{pmatrix}\right]\)

💡 Short Explanation

Multiply every element by scalar 5 \(5\times 0.2=1.0\) \(5\times 0.3=1.5\)

✅ Correct Answer Explanation

Scalar multiplication multiplies each matrix entry by the scalar: \(5\left[\begin{pmatrix}0.2 & 0.3\end{pmatrix}\right]=\left[\begin{pmatrix}1.0 & 1.5\end{pmatrix}\right]\) Thus Option A is correct. Option B changes notation inconsistently, while Options C and D ignore proper scalar multiplication rules.

❓ Why Other Options Are Incorrect

Option B → Numerically close, but matrix entries are inconsistently represented.
Option C → Leaves matrix unchanged instead of scaling by 5.
Option D → Scalar multiplication affects entries proportionally, not by replacing them with scalar values.

🎯 Strategy

Used: Substitution

Application:

Multiply scalar 5 directly with each element of the matrix.

Final Logic:

Element-wise multiplication gives \(\left[1.0 1.5\right]\).

🧠 Memory Trick

"Scalar touches every entry."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Scalar Multiplication

5 Assertion (A): The negative matrix \(-A\) is obtained by taking reciprocals.
Reason (R): \(-A=(-1)A\).

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Negative matrix uses sign change Reciprocals are unrelated \(-A=(-1)A\) defines negative matrix

✅ Correct Answer Explanation

The negative of a matrix is formed by multiplying each element by \(-1\): \(-A=(-1)A\) . It is not obtained by taking reciprocals. Therefore, Assertion (A) is false while Reason (R) is true. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Reason statement is actually correct.
Option B → Assertion is false, not true.
Option C → Since Assertion is false, both statements cannot be true together.

🎯 Strategy

Used: Elimination

Application:

Check the mathematical definition of negative matrix directly.

Final Logic:

Negative means sign reversal, not reciprocal.

🧠 Memory Trick

"Minus flips sign only."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Negative of a Matrix

6 Arrange steps to show \(-(-A)=A\):
1 Multiply scalar inside matrix
2 Identify \(-A\)
3 Negative signs cancel giving \(A\)
4 Apply outer scalar \(-1\)

A 1, 2, 3, 4

B 2, 1, 4, 3

C 2, 4, 1, 3

D 4, 3, 2, 1

💡 Short Explanation

Start with negative matrix Apply outer negative sign Signs cancel to recover original matrix

✅ Correct Answer Explanation

To evaluate: \(-(-A)\) first identify \(-A\), then apply outer scalar \(-1\). Next multiply the scalar inside the matrix. Finally, negative signs cancel and the original matrix \(A\) is obtained. Therefore, the correct order is: 2, 4, 1, 3.

❓ Why Other Options Are Incorrect

Option A → Starts multiplication before identifying the negative matrix.
Option B → Incorrect placement of scalar multiplication step.
Option D → Logical order is reversed completely.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Arrange steps according to the logical flow of evaluating nested negatives.

Final Logic:

Double negatives restore original matrix.

🧠 Memory Trick

"Minus of minus returns plus."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Additive Inverse

7 Let
\(M_{1}=\left[\begin{pmatrix}50 & 60\\ 70 & 80\end{pmatrix}\right],M_{2}=\left[\begin{pmatrix}45 & 55\\ 65 & 75\end{pmatrix}\right]\)
The price drop matrix is defined as:
\(D=M_{1}-M_{2}\)
Find \(D\).

A \(\left[\begin{pmatrix}5 & 5\\ 5 & 5\end{pmatrix}\right]\)

B \(\left[\begin{pmatrix}95 & 115\\ 135 & 155\end{pmatrix}\right]\)

C \(\left[\begin{pmatrix}-5 & -5\\ -5 & -5\end{pmatrix}\right]\)

D \(\left[\begin{pmatrix}50 & 60\\ 70 & 80\end{pmatrix}\right]\)

💡 Short Explanation

Subtract corresponding entries \(50-45=5\) etc. All entries reduce by 5

✅ Correct Answer Explanation

Matrix subtraction is performed element-wise: \(D=\left[\begin{pmatrix}50-45 & 60-55\\ 70-65 & 80-75\end{pmatrix}\right]=\left[\begin{pmatrix}5 & 5\\ 5 & 5\end{pmatrix}\right]\) Thus Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Adds matrices instead of subtracting them.
Option C → Reverses subtraction order incorrectly.
Option D → Leaves original matrix unchanged instead of subtracting.

🎯 Strategy

Used: Substitution

Application:

Subtract corresponding entries directly.

Final Logic:

Element-wise subtraction gives all entries equal to 5.

🧠 Memory Trick

"Subtract same positions."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Subtraction of Matrices

8 Why is
\(A-B=A+(-B)?\)
1. Subtraction is defined as addition of additive inverse
2. Matrix subtraction is a completely different operator
3. Both give element-wise result \(a_{ij}-b_{ij}\)

A 1 only

B 1 and 3

C 2 and 3

D 1, 2, and 3

💡 Short Explanation

Subtraction uses additive inverse Element-wise subtraction remains same Matrix subtraction is not separate fundamentally

✅ Correct Answer Explanation

Matrix subtraction is defined through addition of the additive inverse: \(A-B=A+(-B)\) . Both methods produce identical element-wise results \(a_{ij}-b_{ij}\). Therefore Statements 1 and 3 are correct. Statement 2 is false because subtraction is not an independent operation.

❓ Why Other Options Are Incorrect

Option A → Ignores the valid element-wise interpretation in Statement 3.
Option C → Statement 2 is false since subtraction depends on additive inverse.
Option D → Includes incorrect Statement 2.

🎯 Strategy

Used: Elimination

Application:

Check which statements align with the formal definition of subtraction.

Final Logic:

Subtraction equals addition of inverse.

🧠 Memory Trick

"Subtract = add opposite."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Subtraction of Matrices

9 Let
\(u=\left[\begin{pmatrix}2 & 3\end{pmatrix}\right],v=\left[\begin{pmatrix}4 & 1\end{pmatrix}\right]\)
Which property ensures
\(u+v=v+u?\)

A Associative Law

B Scalar Multiplication

C Commutative Law

D Additive Identity

💡 Short Explanation

Order of addition can change Sum remains same This defines commutativity

✅ Correct Answer Explanation

The commutative law of matrix addition states: \(A+B=B+A\) . Therefore, \(u+v=v+u\) is guaranteed by commutativity. Associativity concerns grouping, scalar multiplication concerns scaling, and additive identity concerns zero matrix properties.

❓ Why Other Options Are Incorrect

Option A → Associativity changes grouping, not order.
Option B → Scalar multiplication is unrelated to matrix addition order.
Option D → Additive identity involves adding zero matrix only.

🎯 Strategy

Used: Option Grouping

Application:

Match the equation with the standard matrix property.

Final Logic:

Changing order without changing sum means commutativity.

🧠 Memory Trick

"Swap order, same sum."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Commutative Property of Addition

10 Let
\(A=\left[\begin{aligned}2\end{aligned}\right],B=\left[\begin{aligned}3\end{aligned}\right],C=\left[\begin{aligned}5\end{aligned}\right]\)
Find the intermediate matrix \(B+C\).
\(B+C=\left[\begin{aligned}8\end{aligned}\right]\)

A \(\left[\begin{aligned}8\end{aligned}\right]\)

B \(\left[\begin{aligned}5\end{aligned}\right]\)

C \(\left[\begin{aligned}10\end{aligned}\right]\)

D \(\left[\begin{aligned}3\end{aligned}\right]\)

💡 Short Explanation

Add scalar entries directly \(3+5=8\) Result is single-entry matrix

✅ Correct Answer Explanation

Since both matrices are \(1\times 1\), addition is simple scalar addition: \(B+C=\left[\begin{aligned}3+5\end{aligned}\right]=\left[\begin{aligned}8\end{aligned}\right]\) Thus Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Represents only one original matrix value.
Option C → Incorrect arithmetic addition.
Option D → Ignores second matrix entry entirely.

🎯 Strategy

Used: Substitution

Application:

Directly add corresponding matrix entries.

Final Logic:

\(3+5=8\)

🧠 Memory Trick

"Add inside entries."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Associative Property of Addition

11 If \(O\) is the zero matrix, what role does it play under matrix addition?

A Additive inverse

B Multiplicative identity

C Scalar multiplier

D Additive identity

💡 Short Explanation

Zero matrix leaves matrices unchanged \(A+O=A\) for all matrices Hence it acts as additive identity

✅ Correct Answer Explanation

The zero matrix acts as the additive identity in matrix algebra because adding it to any matrix leaves the matrix unchanged: \(A+O=A\) . Therefore, Option D is correct. Option A refers to additive inverse, Option B to multiplicative identity, and Option C is unrelated to identity properties.

❓ Why Other Options Are Incorrect

Option A → Additive inverse of \(A\) is \(-A\), not the zero matrix.
Option B → Multiplicative identity is the identity matrix \(I\), satisfying \(AI=IA=A\).
Option C → Scalar multiplication changes matrix entries and is not an identity role.

🎯 Strategy

Used: Elimination

Application:

Use the defining identity property under addition to remove unrelated matrix operations.

Final Logic:

Since \(A+O=A\), zero matrix is the additive identity.

🧠 Memory Trick

"Add zero → Same matrix."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Zero Matrix and Additive Identity

12 Let Monday's cash flow be
\(C=\left[\begin{pmatrix}100 & -50\end{pmatrix}\right]\)
If Tuesday's flow exactly cancels Monday's, what does Tuesday represent?

A Multiplicative inverse

B Additive inverse

C Identity matrix

D Transpose

💡 Short Explanation

Opposite matrix cancels original matrix \(A+(-A)=O\) Tuesday flow balances Monday completely

✅ Correct Answer Explanation

If Tuesday's cash flow exactly cancels Monday's flow, their sum becomes the zero matrix: \(C+(-C)=O\) . Therefore, Tuesday's matrix represents the additive inverse of \(C\). Option B is correct. Other options describe different matrix operations unrelated to cancellation.

❓ Why Other Options Are Incorrect

Option A → Multiplicative inverse relates to matrix inversion, not cancellation under addition.
Option C → Identity matrix leaves multiplication unchanged, not addition.
Option D → Transpose only interchanges rows and columns without cancelling values.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Interpret "exactly cancels" as additive inverse behavior in matrices.

Final Logic:

Cancellation under addition means additive inverse.

🧠 Memory Trick

"Opposite entries cancel."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Negative of a Matrix

📄 Passage

For matrices \(A(m\times n)\)and \(B(n\times p)\), the product \(AB\) is defined and has order \(m\times p\). Matrix multiplication is generally not commutative. The distributive law holds:
\(A(B+C)=AB+AC\)

Which expression correctly represents distributive law?

A \(A(B+C)=AB+BA\)

B \((A+B)C=AC+BC\)

C \(A(B+C)=AB+AC\)

D \(AB+AC=A+B+C\)

💡 Short Explanation

Matrix multiplication distributes over addition Multiply \(A\) separately with \(B\) and \(C\) Add resulting matrices

✅ Correct Answer Explanation

The distributive law for matrices states: \(A(B+C)=AB+AC\) . Matrix \(A\) multiplies each matrix inside the bracket individually. Hence Option C is correct. Option A changes multiplication order incorrectly, Option B represents another distributive form, and Option D is mathematically invalid.

❓ Why Other Options Are Incorrect

Option A → \(AB+BA\) incorrectly changes order; matrix multiplication is generally non-commutative.
Option B → Though valid separately, it does not match the exact expression asked.
Option D → Matrix multiplication cannot disappear into simple addition.

🎯 Strategy

Used: Option Grouping

Application:

Identify the exact standard distributive identity used in NCERT matrix algebra.

Final Logic:

Distribute \(A\) over both matrices:
\(A(B+C)=AB+AC\)

🧠 Memory Trick

"Multiply each inside term."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Distributive Property of Matrix Multiplication

📄 Passage

If a scalar \(k\) multiplies a sum of matrices, which is correct?

A \(k(A+B)=kA+kB\)

B \(k(A+B)=A+kB\)

C \(k(A+B)=kAB\)

D \(k(A+B)=A+B\)

💡 Short Explanation

Scalar multiplication distributes over addition Multiply scalar with each matrix separately Order of matrices remains unchanged

✅ Correct Answer Explanation

Scalar multiplication satisfies the distributive property: \(k(A+B)=kA+kB\) . The scalar multiplies every element of both matrices individually. Therefore, Option A is correct. Other options either omit multiplication correctly or use invalid matrix operations.

❓ Why Other Options Are Incorrect

Option B → Scalar must multiply both matrices, not only one.
Option C → \(AB\) represents matrix multiplication, unrelated to scalar distribution here.
Option D → Scalar multiplication cannot leave the matrix unchanged unless \(k=1\).

🎯 Strategy

Used: Substitution

Application:

Apply the distributive property of scalar multiplication directly.

Final Logic:

Scalar distributes across matrix addition.

🧠 Memory Trick

"Scalar multiplies all terms."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Scalar Multiplication

15 Let cost matrix \(A=(1\times 3)\)and promotion matrix \(B=(3\times 2)\).
Which multiplication gives total cost per city?

A \(AB\)

B \(BA\)

C Both

D Neither

💡 Short Explanation

Inner dimensions must match \(\left(1\times 3)(3\times 2\right)\)is defined Result gives \(1\times 2\) city totals

✅ Correct Answer Explanation

Matrix multiplication is defined when columns of the first matrix equal rows of the second. Here, \(\left(1\times 3)(3\times 2\right)\) is valid and gives a \(1\times 2\) matrix. Hence \(AB\) is defined. But \(\left(3\times 2)(1\times 3\right)\) is invalid since \(2\neq 1\). Therefore, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → \(BA\) is undefined because inner dimensions do not match.
Option C → Only one multiplication order works, not both.
Option D → \(AB\) is clearly defined under matrix multiplication rules.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Check compatibility of inner dimensions before multiplication.

Final Logic:

Since \(3=3\), \(AB\) exists.

🧠 Memory Trick

"Inside numbers must match."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Conditions for Matrix Multiplication

16 Let
\(A=\left[\begin{pmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ 2 & 1 & 0\end{pmatrix}\right],B=\left[\begin{pmatrix}1 & 0 & 2 & 1 & 3\\ 2 & 1 & 0 & 4 & 5\\ 3 & 2 & 1 & 0 & 6\end{pmatrix}\right]\)
Find the element in the 3rd row and 2nd column of the product matrix \(AB\).

A 26

B 34

C 33

D 32

💡 Short Explanation

Use row–column multiplication rule Multiply 3rd row of \(A\) with 2nd column of \(B\) Sum obtained products carefully

✅ Correct Answer Explanation

The 3rd row of \(A\) is: \(\left[7 8 9\right]\) and the 2nd column of \(B\) is: \(\left[\begin{aligned}0\\ 1\\ 2\end{aligned}\right]\) . Therefore, \(7(0)+8(1)+9(2)=0+8+18=26\) So the correct answer should actually be:

❓ Why Other Options Are Incorrect

Option B → Comes from incorrect multiplication or addition during row–column computation.
Option C → Provided answer is mathematically incorrect after proper calculation.
Option D → Obtained from arithmetic error in summation.

🎯 Strategy

Used: Substitution

Application:

Substitute row and column entries directly into multiplication formula.

Final Logic:

\(7(0)+8(1)+9(2)=26\)

🧠 Memory Trick

"Row dot column."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Row–Column Multiplication Rule

17 If \(A\) is \(m\times n\) and \(B\) is \(n\times p\), then
\(c_{ik}=?\)

A \(\sum_{j=1}^{n}\,a_{ij}b_{jk}\)

B \(\sum_{j=1}^{m}\,a_{ij}b_{jk}\)

C \(a_{ik}b_{ik}\)

D \(\sum_{j=1}^{p}\,a_{ij}b_{jk}\)

💡 Short Explanation

Product entries use summation notation Multiply row and column entries Sum over common dimension \(n\)

✅ Correct Answer Explanation

For matrix multiplication, \(c_{ik}=\sum_{j=1}^{n}\,a_{ij}b_{jk}\) where \(n\) is the common dimension. Each entry is obtained by multiplying corresponding row and column elements and then summing them. Hence Option A is correct. Other options use incorrect limits or formulas.

❓ Why Other Options Are Incorrect

Option B → Summation should run through common dimension \(n\), not \(m\).
Option C → Matrix multiplication is not simple element-wise multiplication.
Option D → Upper limit cannot be \(p\) because \(p\) is output column count.

🎯 Strategy

Used: Option Grouping

Application:

Identify the standard NCERT formula for matrix product entries.

Final Logic:

Common dimension \(n\) determines summation range.

🧠 Memory Trick

"Sum across shared dimension."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Product of Two Matrices

18 Let \(X\) be of order \(m\times n\) and \(Y\) be of order \(n\times p\).
Then \(XY\) is of order \(m\times p\).
If \(XY\) is a square matrix, what must be true?

A \(n=p\)

B \(m=p\)

C \(m=n\)

D \(n=m\)

💡 Short Explanation

Square matrix has equal rows and columns \(XY\) has order \(m\times p\) Hence \(m=p\)

✅ Correct Answer Explanation

The product matrix \(XY\) has order: \(m\times p\) . A square matrix requires equal number of rows and columns. Therefore, \(m=p\) must hold. Hence Option B is correct. Other options compare unrelated dimensions.

❓ Why Other Options Are Incorrect

Option A → Equality of \(n\) and \(p\) does not guarantee square result.
Option C → \(m=n\) concerns only first matrix dimensions.
Option D → Equality of \(n\) and \(m\) is unnecessary for square product.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Use output order condition for square matrices.

Final Logic:

Square matrix implies rows = columns:
\(m=p\)

🧠 Memory Trick

"Square → outer dimensions equal."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Order of Product Matrix

19 Let
\(A=\left[\begin{pmatrix}1 & 2\\ 0 & 1\end{pmatrix}\right],B=\left[\begin{pmatrix}2 & 0\\ 1 & 3\end{pmatrix}\right]\)
Compute:
\(AB=\left[\begin{pmatrix}4 & 6\\ 1 & 3\end{pmatrix}\right],BA=\left[\begin{pmatrix}2 & 4\\ 1 & 5\end{pmatrix}\right]\)
Since \(AB\neq BA\), what property is demonstrated?

A Identity existence

B Associativity

C Commutativity

D Non-commutativity

💡 Short Explanation

Changing multiplication order changes result \(AB\neq BA\) Matrix multiplication is generally non-commutative

✅ Correct Answer Explanation

The matrices satisfy: \(AB\neq BA\) . This directly demonstrates the non-commutative nature of matrix multiplication. Hence Option D is correct. Associativity concerns grouping, not order, while commutativity would require equal products.

❓ Why Other Options Are Incorrect

Option A → Identity existence involves identity matrix properties, not unequal products.
Option B → Associativity relates to \(\left(AB)C=A(BC\right)\), not order reversal.
Option C → Commutativity would require \(AB=BA\), which is false here.

🎯 Strategy

Used: Odd One Out

Application:

Identify the property violated when multiplication order changes.

Final Logic:

Unequal products confirm non-commutativity.

🧠 Memory Trick

"Order matters in multiplication."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Non-Commutativity of Matrix Multiplication

20 Let \(A\) be of order \(m\times n\).
For both \(AB\) and \(BA\) to be defined, what must be the order of \(B\)?

A \(n\times m\)

B \(m\times n\)

C \(n\times n\)

D \(m\times m\)

💡 Short Explanation

\(AB\) requires columns of \(A\)=rows of \(B\) \(BA\) requires columns of \(B\)=rows of \(A\) Hence \(B\) must be \(n\times m\)

✅ Correct Answer Explanation

If \(A\) is \(m\times n\), then for \(AB\) to exist, \(B\) must have \(n\) rows. For \(BA\) to exist simultaneously, \(B\) must also have \(m\) columns. Thus \(B\) must be of order: \(n\times m\) . Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Same order does not always guarantee both products exist.
Option C → Only ensures \(AB\) may exist in special cases.
Option D → Does not satisfy both compatibility conditions generally.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Apply multiplication compatibility rules in both directions.

Final Logic:

Both products require:
\(B=n\times m\)

🧠 Memory Trick

"Reverse dimensions for both products."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Conditions for Matrix Multiplication

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