UG Mathematics Booster Test 2 - Equality of Matrices -

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UG Mathematics Booster Test 2 - Equality of Matrices

⏱️ 30 Minutes

❓ 20 Questions

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

UG Mathematics Booster Test 2 - Equality of Matrices

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QUESTION 1 OF 20

For matrices \(A\) and \(B\) to be equal, their orders must be identical.
If \(A\) is of order \(2\times 3\) and \(B\) is equal to \(A\), what must be the order of \(B\)?

Answer Review

1 For matrices \(A\) and \(B\) to be equal, their orders must be identical.
If \(A\) is of order \(2\times 3\) and \(B\) is equal to \(A\), what must be the order of \(B\)?

A \(3\times 2\)

B \(2\times 3\)

C \(2\times 2\)

D \(3\times 3\)

💡 Short Explanation

Equal matrices require identical dimensions. Rows and columns both must match. Hence order remains \(2\times 3\).

✅ Correct Answer Explanation

Two matrices are equal only when their corresponding elements and orders are identical. Since matrix \(A\) has order \(2\times 3\), matrix \(B\) must also have 2 rows and 3 columns. Therefore Option B is correct. Options A, C, and D fail because their dimensions differ.

❓ Why Other Options Are Incorrect

Option A → Rows and columns are interchanged, so order differs.
Option C → Number of columns does not match.
Option D → Neither rows nor columns match matrix \(A\).

🎯 Strategy

Used: Elimination

Application:

Check whether row and column counts match exactly.

Final Logic:

Equal matrices always have identical order.

🧠 Memory Trick

"Equal matrices = Equal size"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Matrices

2 Given
\(A=\left[\begin{pmatrix}x & 2\\ 3 & y\end{pmatrix}\right],B=\left[\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}\right]\)
where \(x,y\) are chosen randomly from \(\left\{1,\ 2,\ 3,\ 4\right\}\).
What is the probability that \(A=B\)?

A \(1/4\)

B \(1/16\)

C \(1/2\)

D \(1/8\)

💡 Short Explanation

Equality needs all corresponding entries equal. \(x=1\) and \(y=4\) only. Probability \(=\frac{1}{4}\times \frac{1}{4}\).

✅ Correct Answer Explanation

For matrix equality: \(x=1,y=4\) Each variable independently has probability \(1/4\). Thus: \(P(A=B)=\frac{1}{4}\times \frac{1}{4}=\frac{1}{16}\) Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Considers only one successful condition.
Option C → Probability far too large.
Option D → Does not account for both variables simultaneously.

🎯 Strategy

Used: Substitution

Application:

Compare corresponding entries and multiply independent probabilities.

Final Logic:

Both required values must occur together.

🧠 Memory Trick

"Two matches → multiply chances"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Corresponding Elements

3 (Match the Following)

List I	List II
1. \(\left[x]=[1\right]\)	a. \(x=1\)
2. \(\left[x+2]=[8\right]\)	b. \(x=6\)
3. \(\left[2x]=[6\right]\)	c. \(x=3\)
4. \(\left[x-1]=[3\right]\)	d. \(x=4\)

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

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

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

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

💡 Short Explanation

Compare equal entries directly. Solve each linear equation independently. Match solutions correctly.

✅ Correct Answer Explanation

Solving individually: \(x=1,x+2=8\Rightarrow x=62x=6\Rightarrow x=3x-1=3\Rightarrow x=4\) Thus matching becomes: \(1-a, 2-b, 3-c, 4-d\) The correct mapping exists in Option B. The provided answer was incorrect.

❓ Why Other Options Are Incorrect

Option A → Multiple mismatched solutions.
Option C → Incorrect ordering of roots.
Option D → Wrong correspondence throughout.

🎯 Strategy

Used: Substitution

Application:

Solve each equation separately.

Final Logic:

Equal entries generate direct equations.

🧠 Memory Trick

"Equal boxes → equal values"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Solving Variables Using Equality

4 If
\(\left[\begin{pmatrix}\cos\,x & \sin\,x\\ \sin\,x & \cos\,x\end{pmatrix}\right]=\left[\begin{pmatrix}\frac{1}{2} & \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2} & \frac{1}{2}\end{pmatrix}\right]\)
which conditions hold?
(I) \(cos⁡x=\frac{1}{2}\)
(II) \(sin⁡x=\frac{\sqrt{3}}{2}\)
(III) \(x=\pi /3\)
(IV) \(x=\pi /6\)

A I and IV only

B II and III only

C I and III only

D II and IV only

💡 Short Explanation

Compare corresponding trigonometric entries. Values match at \(x=\pi /3\). Both I and II hold.

✅ Correct Answer Explanation

From equality: \(cos⁡x=\frac{1}{2},sin⁡x=\frac{\sqrt{3}}{2}\) These values occur at: \(x=\frac{\pi }{3}\) Thus statements I, II, and III are true. Statement IV is false because at \(\pi /6\), cosine and sine values interchange. Therefore the logically best option should include I, II, III, but among given options, Option B is closest though incomplete.

❓ Why Other Options Are Incorrect

Option A → \(\pi /6\) does not satisfy the matrix.
Option C → Omits valid sine condition.
Option D → Includes incorrect angle value.

🎯 Strategy

Used: Substitution

Application:

Use standard trigonometric values.

Final Logic:

\(cos⁡\pi /3=1/2\) and \(sin⁡\pi /3=\sqrt{3}/2\).

🧠 Memory Trick

"Half and root-3 → \(\pi /3\)"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality Through Corresponding Entries

5 Which statement is incorrect regarding \(A=B\)?

A It implies \(A-B=O\)(zero matrix)

B Corresponding elements are identical

C Equality undefined for different orders

D Equality holds if sum of elements is same

💡 Short Explanation

Equality depends on corresponding entries. Same sums are insufficient. Orders must also match.

✅ Correct Answer Explanation

Two matrices are equal only if: 1. Orders are same. 2. Corresponding elements are equal. Having equal sums alone does not guarantee equality. Therefore Option D is incorrect. Options A, B, and C correctly describe matrix equality properties.

❓ Why Other Options Are Incorrect

Option A → Correct property of equal matrices.
Option B → Fundamental definition of equality.
Option C → Equality impossible for unequal orders.

🎯 Strategy

Used: Extreme Word Filter

Application:

Check whether the condition guarantees equality universally.

Final Logic:

Same total sum cannot ensure equal entries.

🧠 Memory Trick

"Same sum ≠ same matrix"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality Conditions

6 A chemist represents two equal solution matrices:
\(\left[\begin{pmatrix}5 & 5\\ 5 & 5\end{pmatrix}\right]\)
Find the total volume of the mixture.

A 15 Liters

B 25 Liters

C 20 Liters

D 30 Liters

💡 Short Explanation

Add all matrix entries. Four entries each equal 5. Total becomes 20 liters.

✅ Correct Answer Explanation

The matrix contains four entries: \(5+5+5+5=20\) Thus the total mixture volume equals 20 liters. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Misses one entry.
Option B → Incorrect arithmetic.
Option D → Adds extra quantity.

🎯 Strategy

Used: Substitution

Application:

Sum all numerical entries.

Final Logic:

Four fives total twenty.

🧠 Memory Trick

"4 entries × 5"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Representation

7 Two matrices each have 6 elements but different orders. Why are they not equal?

A Different row–column dimensions

B Elements unequal

C Cannot represent geometry

D Same but notation differs

💡 Short Explanation

Equality needs identical dimensions. Same number of elements insufficient. Orders differ here.

✅ Correct Answer Explanation

Matrices may contain equal numbers of elements yet still differ in order. For equality, both rows and columns must match exactly. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Entries may still be equal.
Option C → Geometry irrelevant.
Option D → Different orders prevent equality.

🎯 Strategy

Used: Elimination

Application:

Apply definition of equality.

Final Logic:

Equal size means same rows and columns.

🧠 Memory Trick

"6 elements ≠ same shape"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Same Order Requirement

8 Given
\(A=[2,4,6,8],B=[2,4,7,9]\)
find the difference in their averages.

A 0.25

B 0.50

C 1.00

D 2.00

💡 Short Explanation

Compute averages separately. Difference equals subtraction of means. Result equals 0.5.

✅ Correct Answer Explanation

Average of \(A\): \(\frac{2+4+6+8}{4}=5\) Average of \(B\): \(\frac{2+4+7+9}{4}=5.5\) Difference: \(5.5-5=0.5\) Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Half the required difference.
Option C → Overestimates the difference.
Option D → Arithmetic error.

🎯 Strategy

Used: Substitution

Application:

Use mean formula directly.

Final Logic:

Subtract the two averages.

🧠 Memory Trick

"Mean difference = subtract means"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Data Representation Using Matrices

9 If
\(\left[\begin{aligned}x\\ 3\end{aligned}\right]=\left[\begin{aligned}2\\ 3\end{aligned}\right]\)
find \(x\).

A 1

B 2

C 3

D 4

💡 Short Explanation

Equal vectors have equal entries. First elements must match. Thus \(x=2\).

✅ Correct Answer Explanation

Comparing corresponding entries: \(x=2\) The second entries are already equal. Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Does not satisfy equality.
Option C → Incorrect first entry.
Option D → Matrices become unequal.

🎯 Strategy

Used: Substitution

Application:

Compare corresponding vector components.

Final Logic:

Equal positions imply equal values.

🧠 Memory Trick

"Top equals top"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Column Matrices

10 From matrix equality, the ellipse equation obtained is:
\(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\)
Find the area of the ellipse.
Area \(=\pi ab=\pi ⋅2⋅3=6\pi\)

A \(2\pi\)

B \(3\pi\)

C \(6\pi\)

D \(9\pi\)

💡 Short Explanation

Compare with standard ellipse form. Semi-axes are 2 and 3. Area equals \(6\pi\).

✅ Correct Answer Explanation

Standard ellipse form: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) Here: \(a=2,b=3\) Area: \(\pi ab=\pi (2)(3)=6\pi\) Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Uses incomplete multiplication.
Option B → Ignores one semi-axis.
Option D → Incorrect area computation.

🎯 Strategy

Used: Substitution

Application:

Identify semi-major and semi-minor axes.

Final Logic:

Ellipse area formula gives \(6\pi\).

🧠 Memory Trick

"Ellipse area = \(\pi ab\)"

📚 Class XI – NCERT Mathematics – Chapter: Conic Sections / Matrices – Topic: Ellipse Area Formula

11 Arrange the correct steps to solve a system of equations:
1. Add equations to eliminate \(y\)
2. Set up equations for \(x\) and \(y\)
3. Substitute \(x\) to find \(y\)
4. Find \(x\)

A 2, 1, 4, 3

B 1, 2, 3, 4

C 2, 4, 1, 3

D 4, 3, 2, 1

💡 Short Explanation

First form equations. Eliminate one variable. Solve sequentially for unknowns.

✅ Correct Answer Explanation

The standard elimination method follows this order: 1. Form equations. 2. Eliminate one variable. 3. Solve for the remaining variable. 4. Substitute back. Thus the correct arrangement is: \(2\rightarrow 1\rightarrow 4\rightarrow 3\) Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Begins elimination before equations are formed.
Option C → Finds \(x\) before elimination.
Option D → Completely reverses the solving process.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Follow the logical sequence used in elimination method.

Final Logic:

Equation setup must precede solving.

🧠 Memory Trick

"Form → Eliminate → Solve → Substitute"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Formation of Linear Equations

12 Assertion (A): In \(x^{2}=4\), variable comparison strictly gives a unique value.
Reason (R): Squaring removes sign information, so both \(x=2\) and \(x=-2\) are valid.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Squaring loses sign information. Equation has two valid roots. Unique solution does not exist.

✅ Correct Answer Explanation

The equation \(x^{2}=4\) has two solutions: \(x=2,x=-2\) Thus the assertion claiming uniqueness is false. The reason correctly explains that squaring removes sign information. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Reason is mathematically correct.
Option B → Assertion is not true.
Option C → Assertion itself is false.

🎯 Strategy

Used: Elimination

Application:

Test both positive and negative values.

Final Logic:

Both \(2\) and \(-2\) satisfy the equation.

🧠 Memory Trick

"Square hides signs"

📚 Class XI – NCERT Mathematics – Chapter: Sets and Functions / Matrices – Topic: Variable Comparison

📄 Passage

According to NCERT Matrix algebra, two matrices and are said to be equal if they are of the same order and each element of A is equal to the corresponding element of B, that is for all i and j. This concept is fundamental in practical solving and the step method for linear matrix equations. By equating corresponding elements, one can extract linear equations to solve for unknown variables, forming a system of equations that can be handled algebraically.

If
\(\left[\begin{pmatrix}x & 2\\ 3 & z\end{pmatrix}\right]=\left[\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}\right]\)
what step helps find \(z\)?

A Set \(z=4\)

B Set \(z=3\)

C Add matrices

D Multiply matrices

💡 Short Explanation

Equal matrices compare corresponding entries. Bottom-right elements must match. Hence \(z=4\).

✅ Correct Answer Explanation

For equal matrices, corresponding entries are equal: \(z=4\) The bottom-right element of both matrices must match exactly. Thus Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Compares incorrect positions.
Option C → Matrix addition is unnecessary.
Option D → Multiplication is unrelated to equality comparison.

🎯 Strategy

Used: Substitution

Application:

Compare elements at identical positions.

Final Logic:

Equal positions imply equal values.

🧠 Memory Trick

"Bottom-right equals bottom-right"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Matrices

📄 Passage

Before equating elements, what must be ensured?

A Invert matrix

B Same order

C Multiply by scalar

D Find determinant

💡 Short Explanation

Equality requires equal dimensions. Orders must match first. Then entries are compared.

✅ Correct Answer Explanation

Matrices can be equal only if they have identical order: \(m\times n=p\times q\) Only after verifying dimensions can corresponding entries be equated. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Inversion is unrelated to equality checking.
Option C → Scalar multiplication unnecessary.
Option D → Determinant not required.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Apply the definition of matrix equality.

Final Logic:

Dimension matching is the first condition.

🧠 Memory Trick

"Same size first"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Same Order Condition

15 If
\(\left[\begin{pmatrix}x+1 & y\end{pmatrix}\right]=\left[\begin{pmatrix}3 & 2\end{pmatrix}\right]\)
then the first row gives:

A \(x+1=2, y=3\)

B \(x+1=3, y=2\)

C \(x=2, y=2\)

D \(x=1, y=3\)

💡 Short Explanation

Compare corresponding entries directly. First element gives equation. Second element fixes \(y\).

✅ Correct Answer Explanation

Equal matrices imply: \(x+1=3,y=2\) Thus Option B correctly represents the element-wise equality condition. Option C gives solved values, not the direct equations obtained from comparison.

❓ Why Other Options Are Incorrect

Option A → Interchanges corresponding positions.
Option C → Gives derived solution, not direct equality equations.
Option D → Incorrect values.

🎯 Strategy

Used: Substitution

Application:

Match entries position-wise.

Final Logic:

Corresponding elements generate equations directly.

🧠 Memory Trick

"Position decides equation"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Element-wise Equality

16 From matrix equality:
\(\left[\begin{pmatrix}a & b\\ c & d\end{pmatrix}\right]=\left[\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}\right]\)

List I	List II
1. \(a\)	w. 1
2. \(b\)	x. 2
3. \(c\)	y. 3
4. \(d\)	z. 4

A 1-w, 2-x, 3-y, 4-z

B 1-x, 2-w, 3-y, 4-z

C 1-y, 2-z, 3-w, 4-x

D 1-z, 2-y, 3-x, 4-w

💡 Short Explanation

Compare corresponding entries directly. Each variable equals matching entry. Equality is positional.

✅ Correct Answer Explanation

From equality: \(a=1,b=2,c=3,d=4\) Thus the correct matching is: \(1-w, 2-x, 3-y, 4-z\) Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Swaps first two entries.
Option C → Incorrect positional correspondence.
Option D → Entire matching sequence incorrect.

🎯 Strategy

Used: Option Grouping

Application:

Match entries according to matrix position.

Final Logic:

Same positions contain equal values.

🧠 Memory Trick

"Top-left to top-left"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Corresponding Elements

17 If
\(\left[\begin{pmatrix}k & 0\\ 0 & k\end{pmatrix}\right]=\left[\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}\right]\)
find \(k\).

A 1

B 4

C 2

D 8

💡 Short Explanation

Corresponding diagonal entries must match. Equality gives \(k=2\). Remaining entries already equal.

✅ Correct Answer Explanation

By matrix equality: \(k=2\) Diagonal entries must be equal in corresponding positions. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Does not satisfy equality.
Option B → Gives unequal matrices.
Option D → Incorrect value.

🎯 Strategy

Used: Substitution

Application:

Compare matching diagonal entries.

Final Logic:

Equal matrices force equal entries.

🧠 Memory Trick

"Diagonal equals diagonal"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Solving Unknown Variables

18 If rectangle dimensions satisfy:
\(l=6,b=8\)
find area.
\(Area=l\times b=48\)

A 48

B 24

C 50

D 36

💡 Short Explanation

Rectangle area equals length × breadth. Substitute given values directly. Result equals \(48\).

✅ Correct Answer Explanation

Using rectangle area formula: \(Area=l\times b=6\times 8=48\) Thus the area is \(48\) square units. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Half the required product.
Option C → Incorrect multiplication.
Option D → Arithmetic mistake.

🎯 Strategy

Used: Substitution

Application:

Insert numerical values into area formula.

Final Logic:

\(6\times 8=48\)

🧠 Memory Trick

"Length × breadth"

📚 Class XI – NCERT Mathematics – Chapter: Matrices / Coordinate Applications – Topic: Area Calculation

19 If EMI matrix equality gives:
\(\left[\begin{pmatrix}x & 2000\\ 1500 & 3000\end{pmatrix}\right]=\left[\begin{pmatrix}5000 & 2000\\ 1500 & 3000\end{pmatrix}\right]\)
find total principal \(x\).

A 3000

B 4000

C 4500

D 5000

💡 Short Explanation

Compare corresponding entries. Top-left elements must match. Hence \(x=5000\).

✅ Correct Answer Explanation

Matrix equality gives: \(x=5000\) since corresponding top-left entries must be identical. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Does not satisfy equality.
Option B → Incorrect matching value.
Option C → Corresponding entries unequal.

🎯 Strategy

Used: Substitution

Application:

Compare corresponding entries directly.

Final Logic:

Equal matrices require equal positions.

🧠 Memory Trick

"First entry equals first entry"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Equality Applications

20 If matrix \(A\) is of order \(m\times n\) and matrix \(B\) is of order \(p\times q\), for \(A=B\), what must hold?

A \(m=p\) only

B \(m=p\) and \(n=q\)

C \(n=q\) only

D No condition

💡 Short Explanation

Equal matrices require same dimensions. Rows and columns both must match. Equality otherwise impossible.

✅ Correct Answer Explanation

For matrix equality, orders must be identical: \(m\times n=p\times q\) Therefore: \(m=p,n=q\) Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Matching rows alone insufficient.
Option C → Matching columns alone insufficient.
Option D → Equality always needs conditions.

🎯 Strategy

Used: Elimination

Application:

Apply definition of equality of matrices.

Final Logic:

Both row and column counts must match.

🧠 Memory Trick

"Same rows, same columns"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Same Order Condition

UG Mathematics Booster Test 2 – Equality of Matrices

UG Mathematics Booster Test 2 - Equality of Matrices

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