Matrices and Determinants | MCQs | Set -01

Class 12 Mathematics - Matrices & Determinants

Important & Smart MCQs Marks · 1M

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📗 Top 50 Most Important MCQs

50 Q&A

1 mark1

If \( A \) is a square matrix such that \( A^2 = I \), then \( (A - I)^3 + (A + I)^3 - 7A \) is equal to:

1. \( A \)
2. \( I - A \)
3. \( I + A \)
4. \( 3A \)

Solution: Option (A)

\( \because (A - I)^3 = A^3 - 3A^2I + 3AI^2 - I^3 \)
\( \Rightarrow (A - I)^3 = A(A^2) - 3(I)I + 3A(I) - I \)
\( \Rightarrow (A - I)^3 = A(I) - 3I + 3A - I \)
\( \Rightarrow (A - I)^3 = 4A - 4I \)

Similarly, \( (A + I)^3 = A^3 + 3A^2I + 3AI^2 + I^3 \)
\( \Rightarrow (A + I)^3 = A + 3I + 3A + I \)
\( \Rightarrow (A + I)^3 = 4A + 4I \)

\( \therefore (A - I)^3 + (A + I)^3 - 7A = (4A - 4I) + (4A + 4I) - 7A \)
\( \Rightarrow 8A - 7A \)
\( \Rightarrow A \)

1 mark2

If \( A \) is a \( 3 \times 3 \) non-singular matrix and \( |A| = 4 \), then the value of \( |\text{adj } A| \) is:

1. 4
2. 16
3. 64
4. 8

Solution: Option (B)

\( \because |\text{adj } A| = |A|^{n-1} \)
Here, order \( n = 3 \).
\( \Rightarrow |\text{adj } A| = (4)^{3-1} \)
\( \Rightarrow |\text{adj } A| = 4^2 \)
\( \therefore |\text{adj } A| = 16 \)

1 mark3

For any \( 2 \times 2 \) square matrix \( A \), if \( A(\text{adj } A) = \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix} \), then the value of \( |A| \) is:

1. 20
2. 100
3. 10
4. 0

Solution: Option (C)

By standard property, \( A(\text{adj } A) = |A|I \)
\( \Rightarrow |A| \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix} \)
\( \Rightarrow \begin{bmatrix} |A| & 0 \\ 0 & |A| \end{bmatrix} = \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix} \)
\( \therefore |A| = 10 \)

1 mark4

If \( A \) and \( B \) are symmetric matrices of the same order, then \( (AB - BA) \) is a:

1. Symmetric matrix
2. Skew-symmetric matrix
3. Zero matrix
4. Identity matrix

Solution: Option (B)

\( \because (AB - BA)' = (AB)' - (BA)' \)
\( \Rightarrow (AB - BA)' = B'A' - A'B' \)
Given \( A \) and \( B \) are symmetric, so \( A' = A \) and \( B' = B \).
\( \Rightarrow (AB - BA)' = BA - AB \)
\( \Rightarrow (AB - BA)' = -(AB - BA) \)
\( \therefore \) It is a skew-symmetric matrix.

1 mark5

Let \( A \) be a square matrix of order 3. If \( |A| = 5 \), find the value of \( |2A| \).

1. 10
2. 40
3. 20
4. 125

Solution: Option (B)

\( \because |kA| = k^n |A| \)
Here, \( k = 2 \) and \( n = 3 \).
\( \Rightarrow |2A| = 2^3 \times |A| \)
\( \Rightarrow |2A| = 8 \times 5 \)
\( \therefore |2A| = 40 \)

1 mark6

The value of a determinant of a skew-symmetric matrix of odd order is always:

1. 1
2. -1
3. 0
4. Depends on the elements

Solution: Option (C)

\( \because A^T = -A \)
\( \Rightarrow |A^T| = |-A| \)
\( \Rightarrow |A| = (-1)^n |A| \)
For odd order, \( n \) is odd, so \( (-1)^n = -1 \).
\( \Rightarrow |A| = -|A| \)
\( \Rightarrow 2|A| = 0 \)
\( \therefore |A| = 0 \)

1 mark7

If \( A \) is an orthogonal matrix, then the determinant of \( A \) is:

1. 0
2. \( \pm 1 \)
3. 2
4. None of these

Solution: Option (B)

For an orthogonal matrix, \( AA^T = I \).
\( \Rightarrow |AA^T| = |I| \)
\( \Rightarrow |A||A^T| = 1 \)
\( \Rightarrow |A|^2 = 1 \)
\( \therefore |A| = \pm 1 \)

1 mark8

If \( A \) is an invertible matrix of order 2, then \( \det(A^{-1}) \) equals:

1. \( \det(A) \)
2. \( \frac{1}{\det(A)} \)
3. 1
4. 0

Solution: Option (B)

\( \because A A^{-1} = I \)
\( \Rightarrow |A A^{-1}| = |I| \)
\( \Rightarrow |A| \cdot |A^{-1}| = 1 \)
\( \therefore |A^{-1}| = \frac{1}{|A|} \)

1 mark9

The maximum number of distinct elements in a symmetric matrix of order \( n \) is:

1. \( n^2 \)
2. \( \frac{n(n-1)}{2} \)
3. \( \frac{n(n+1)}{2} \)
4. \( n \)

Solution: Option (C)

In a symmetric matrix, elements below the principal diagonal are identical to those above it.
Number of elements on the principal diagonal = \( n \).
Number of elements above the diagonal = \( \frac{n^2 - n}{2} \).
\( \Rightarrow \text{Total distinct} = n + \frac{n^2 - n}{2} \)
\( \Rightarrow \text{Total distinct} = \frac{2n + n^2 - n}{2} \)
\( \therefore \text{Total distinct} = \frac{n(n+1)}{2} \)

1 mark10

If the system of equations \( AX = B \) has no solution, then the system is called:

1. Consistent
2. Inconsistent
3. Dependent
4. Trivial

Solution: Option (B)

A system of linear equations is called inconsistent if it possesses no solution.
\( \therefore \) Inconsistent.

1 mark11

If \( A \) and \( B \) are invertible matrices of the same order, then \( (AB)^{-1} \) is equal to:

1. \( A^{-1} B^{-1} \)
2. \( B^{-1} A^{-1} \)
3. \( AB \)
4. \( BA \)

Solution: Option (B)

By the reversal law for the inverse of a product of matrices:
\( \therefore (AB)^{-1} = B^{-1} A^{-1} \)

1 mark12

What is the condition for a square matrix \( A \) to be singular?

1. \( |A| \neq 0 \)
2. \( |A| = 1 \)
3. \( |A| = 0 \)
4. \( |A| = -1 \)

Solution: Option (C)

A square matrix is said to be singular if its determinant is zero.
\( \therefore |A| = 0 \)

1 mark13

If \( A = \left[a_{ij}\right] \) is a scalar matrix of order \( n \times n \) such that \( a_{ii} = k \), then \( |A| \) is:

1. \( k \)
2. \( n^k \)
3. \( k^n \)
4. \( nk \)

Solution: Option (C)

A scalar matrix is a diagonal matrix where all diagonal elements are \( k \).
\( \because \) Determinant of a diagonal matrix is the product of its diagonal elements.
\( \Rightarrow |A| = k \times k \times \dots \text{ (n times)} \)
\( \therefore |A| = k^n \)

1 mark14

Let \( \Delta = \left|\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right| \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), then value of \( a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} \) is:

1. \( \Delta \)
2. 0
3. 1
4. \( -\Delta \)

Solution: Option (B)

If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero.
\( \therefore a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0 \)

1 mark15

If the points \( (x_1, y_1), (x_2, y_2), \) and \( (x_3, y_3) \) are collinear, then the area of the triangle formed by them using determinant is:

1. 1
2. 0
3. -1
4. Infinity

Solution: Option (B)

For collinear points, the triangle formed has zero area.
\( \therefore \Delta = 0 \)

1 mark16

If matrix \( A \) is both symmetric and skew-symmetric, then:

1. \( A \) is a diagonal matrix
2. \( A \) is a zero matrix
3. \( A \) is an identity matrix
4. \( A \) is an orthogonal matrix

Solution: Option (B)

Since \( A \) is symmetric, \( A' = A \).
Since \( A \) is skew-symmetric, \( A' = -A \).
\( \Rightarrow A = -A \)
\( \Rightarrow 2A = 0 \)
\( \therefore A = 0 \) (Zero Matrix)

1 mark17

If \( A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \), then \( A - A' \) is a:

1. Null Matrix
2. Symmetric Matrix
3. Skew-Symmetric Matrix
4. Scalar Matrix

Solution: Option (C)

For any square matrix \( A \), the matrix \( (A - A') \) is always a skew-symmetric matrix.
Proof: Let \( P = A - A' \).
\( \Rightarrow P' = (A - A')' = A' - (A')' \)
\( \Rightarrow P' = A' - A = -(A - A') \)
\( \Rightarrow P' = -P \)
\( \therefore \) Skew-symmetric.

1 mark18

If \( A \) is a matrix of order \( 3 \times 4 \) and \( B \) is a matrix of order \( 4 \times 5 \), then the number of columns in the matrix \( AB \) is:

1. 3
2. 4
3. 5
4. 12

Solution: Option (C)

If order of \( A \) is \( m \times n \) and order of \( B \) is \( n \times p \), then order of \( AB \) is \( m \times p \).
\( \Rightarrow \) Order of \( AB \) is \( 3 \times 5 \).
\( \therefore \) Number of columns is 5.

1 mark19

The number of all possible matrices of order \( 2 \times 2 \) with each entry 0 or 1 is:

1. 8
2. 16
3. 64
4. 4

Solution: Option (B)

Order \( 2 \times 2 \) means there are 4 elements.
Each element can be filled in 2 ways (either 0 or 1).
\( \Rightarrow \text{Total possibilities} = 2 \times 2 \times 2 \times 2 \)
\( \therefore \text{Total} = 2^4 = 16 \)

1 mark20

If \( A \) is a diagonal matrix, then \( A^{-1} \) (if it exists) is:

1. Symmetric matrix
2. Skew-symmetric matrix
3. Diagonal matrix
4. Scalar matrix

Solution: Option (C)

The inverse of a diagonal matrix is formed by taking the reciprocal of its non-zero diagonal elements.
\( \therefore \) It remains a diagonal matrix.

1 mark21

If a matrix has 6 elements, what are the possible orders it can have?

1. 2
2. 4
3. 6
4. 3

Solution: Option (B)

The number of elements is 6.
Possible ordered pairs whose product is 6: \( (1, 6), (6, 1), (2, 3), (3, 2) \).
\( \therefore \) There are 4 possible orders.

1 mark22

A system of linear equations \( AX = B \) has infinite solutions if:

1. \( |A| \neq 0 \)
2. \( |A| = 0 \) and \( (\text{adj } A)B \neq O \)
3. \( |A| = 0 \) and \( (\text{adj } A)B = O \)
4. \( |A| = 1 \)

Solution: Option (C)

For infinite solutions, the system must be consistent with infinitely many possibilities.
This occurs when \( |A| = 0 \) and \( (\text{adj } A)B = O \).
\( \therefore \) Option (C) is correct.

1 mark23

If \( A = \left[\begin{array}{cc} x & 2 \\ 2 & x \end{array}\right] \) and \( |A| = 0 \), then the value of \( x \) is:

1. 2
2. -2
3. \( \pm 2 \)
4. 0

Solution: Option (C)

\( \because |A| = 0 \)
\( \Rightarrow x(x) - 2(2) = 0 \)
\( \Rightarrow x^2 - 4 = 0 \)
\( \Rightarrow x^2 = 4 \)
\( \therefore x = \pm 2 \)

1 mark24

If \( A \) is an idempotent matrix, then:

1. \( A^2 = I \)
2. \( A^2 = O \)
3. \( A^2 = A \)
4. \( A^3 = I \)

Solution: Option (C)

By definition, a square matrix is called idempotent if its square equals the matrix itself.
\( \therefore A^2 = A \)

1 mark25

If any two rows (or columns) of a determinant are identical, then the value of the determinant is:

1. 1
2. -1
3. 0
4. Cannot be determined

Solution: Option (C)

By the properties of determinants, if two rows or columns are identical, the determinant evaluates to zero.
\( \therefore \Delta = 0 \)

1 mark26

For a square matrix \( A \), the value of \( \text{Trace}(A) \) is equal to:

1. Product of diagonal elements
2. Sum of diagonal elements
3. Sum of all elements
4. Determinant of \( A \)

Solution: Option (B)

The trace of a square matrix is defined as the sum of its principal diagonal elements.
\( \therefore \text{Tr}(A) = \sum a_{ii} \)

1 mark27

If \( A \) is a matrix such that \( A^3 = I \), then \( A^{-1} \) is equal to:

1. \( A \)
2. \( A^2 \)
3. \( I \)
4. Does not exist

Solution: Option (B)

\( \because A^3 = I \)
Multiplying both sides by \( A^{-1} \):
\( \Rightarrow A^{-1} A^3 = A^{-1} I \)
\( \Rightarrow A^2 = A^{-1} \)
\( \therefore A^{-1} = A^2 \)

1 mark28

The value of \( \left|\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right| \) is:

1. 0
2. 1
3. -1
4. \( \cos 2\theta \)

Solution: Option (B)

\( \because \Delta = (\cos \theta)(\cos \theta) - (-\sin \theta)(\sin \theta) \)
\( \Rightarrow \Delta = \cos^2 \theta + \sin^2 \theta \)
\( \therefore \Delta = 1 \)

1 mark29

If \( |A| = 0 \), then the inverse of matrix \( A \):

1. Is a null matrix
2. Is an identity matrix
3. Does not exist
4. Is equal to adjoint \( A \)

Solution: Option (C)

\( \because A^{-1} = \frac{\text{adj } A}{|A|} \)
If \( |A| = 0 \), the expression is undefined.
\( \therefore \) Inverse does not exist (Matrix is singular).

1 mark30

Let \( A \) and \( B \) be square matrices of the same order. Then \( |AB| \) is equal to:

1. \( |A| + |B| \)
2. \( |A| - |B| \)
3. \( |A||B| \)
4. \( |A| / |B| \)

Solution: Option (C)

By the multiplicative property of determinants:
\( \therefore |AB| = |A| |B| \)

1 mark31

If \( A = \left[a_{ij}\right]_{m \times n} \) is a square matrix, then:

1. \( m < n \)
2. \( m > n \)
3. \( m = n \)
4. None of these

Solution: Option (C)

A matrix is defined as a square matrix if the number of rows is exactly equal to the number of columns.
\( \therefore m = n \)

1 mark32

Every square matrix can be uniquely expressed as the sum of:

1. Two diagonal matrices
2. A symmetric and a skew-symmetric matrix
3. An identity and a null matrix
4. Two singular matrices

Solution: Option (B)

For any square matrix \( A \):
\( \Rightarrow A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A') \)
Where \( \frac{1}{2}(A + A') \) is symmetric and \( \frac{1}{2}(A - A') \) is skew-symmetric.
\( \therefore \) Option (B) is correct.

1 mark33

If \( (AB)' = B' A' \), this property is known as:

1. Associative Law
2. Commutative Law
3. Reversal Law
4. Distributive Law

Solution: Option (C)

The transpose of the product of two matrices is the product of their transposes taken in the reverse order.
\( \therefore \) It is called the Reversal Law.

1 mark34

The value of the determinant \( \left|\begin{array}{ccc} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{array}\right| \) is:

1. \( a+b+c \)
2. 0
3. 1
4. \( abc \)

Solution: Option (B)

Applying column operation \( C_3 \to C_3 + C_2 \):
\( \Rightarrow \Delta = \left|\begin{array}{ccc} 1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c \end{array}\right| \)
Taking \( (a+b+c) \) common from \( C_3 \):
\( \Rightarrow \Delta = (a+b+c) \left|\begin{array}{ccc} 1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1 \end{array}\right| \)
Since \( C_1 \) and \( C_3 \) are identical, the determinant is 0.
\( \therefore \Delta = 0 \)

1 mark35

If \( A = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \), where \( i^2 = -1 \), then \( A^2 \) is equal to:

1. \( I \)
2. \( -I \)
3. \( A \)
4. \( -A \)

Solution: Option (B)

\( \because A^2 = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \)
\( \Rightarrow A^2 = \begin{bmatrix} i^2 & 0 \\ 0 & i^2 \end{bmatrix} \)
\( \Rightarrow A^2 = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \)
\( \Rightarrow A^2 = -1 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
\( \therefore A^2 = -I \)

1 mark36

The minor of an element \( a_{ij} \) of a determinant of order \( n \) is a determinant of order:

1. \( n \)
2. \( n+1 \)
3. \( n-1 \)
4. \( n^2 \)

Solution: Option (C)

The minor of an element is obtained by deleting the row and column containing that element.
\( \therefore \) The order reduces by 1, making it \( n-1 \).

1 mark37

If \( |\text{adj}(\text{adj } A)| = |A|^4 \) for a square matrix \( A \), what is the order of matrix \( A \)?

1. 2
2. 3
3. 4
4. 5

Solution: Option (B)

\( \because |\text{adj}(\text{adj } A)| = |A|^{(n-1)^2} \)
Given \( |A|^{(n-1)^2} = |A|^4 \).
\( \Rightarrow (n-1)^2 = 4 \)
\( \Rightarrow n-1 = 2 \)
\( \therefore n = 3 \)

1 mark38

Which of the following is NOT a valid property of matrix multiplication?

1. \( A(BC) = (AB)C \)
2. \( A(B+C) = AB + AC \)
3. \( AB = BA \) (always)
4. \( AI = IA = A \)

Solution: Option (C)

Matrix multiplication is not commutative in general.
\( \therefore AB = BA \) is not always true.

1 mark39

If a matrix has only one row, it is called a:

1. Column matrix
2. Row matrix
3. Square matrix
4. Singleton matrix

Solution: Option (B)

By definition, a matrix with exactly one row is called a row matrix.
\( \therefore \) Row matrix.

1 mark40

If \( A \) is an upper triangular matrix, then all elements below the principal diagonal are:

1. 1
2. Equal to the diagonal elements
3. 0
4. Negative

Solution: Option (C)

In an upper triangular matrix, \( a_{ij} = 0 \) for all \( i > j \).
\( \therefore \) All elements below the principal diagonal are 0.

1 mark41

If matrix \( A \) is nilpotent of index 2, then:

1. \( A^2 = I \)
2. \( A^2 = A \)
3. \( A^2 = O \)
4. \( A = O \)

Solution: Option (C)

A matrix is nilpotent of index \( k \) if \( A^k = O \) and \( A^{k-1} \neq O \).
\( \therefore A^2 = O \)

1 mark42

The value of \( \left|\begin{array}{cc} x & y \\ x & y \end{array}\right| \) is:

1. \( x^2 - y^2 \)
2. \( 2xy \)
3. 0
4. \( x - y \)

Solution: Option (C)

\( \because \Delta = (x)(y) - (y)(x) \)
\( \Rightarrow \Delta = xy - xy \)
\( \therefore \Delta = 0 \)

1 mark43

If \( A \) and \( B \) are square matrices such that \( AB = I \) and \( BA = I \), then \( B \) is called the:

1. Transpose of \( A \)
2. Adjoint of \( A \)
3. Inverse of \( A \)
4. Determinant of \( A \)

Solution: Option (C)

By definition of an invertible matrix, if their product yields the identity matrix, they are inverses of each other.
\( \therefore \) Inverse of \( A \).

1 mark44

If the rows of a determinant are changed into columns and columns into rows, the value of the determinant:

1. Becomes zero
2. Changes its sign
3. Remains unchanged
4. Is squared

Solution: Option (C)

By the first property of determinants, the determinant of a matrix and its transpose are identical.
\( \therefore \) Remains unchanged.

1 mark45

If \( A \) is a symmetric matrix, then \( A^{-1} \) (if it exists) is:

1. Symmetric
2. Skew-symmetric
3. Diagonal
4. Scalar

Solution: Option (A)

\( \because (A^{-1})' = (A')^{-1} \)
Given \( A \) is symmetric, \( A' = A \).
\( \Rightarrow (A^{-1})' = A^{-1} \)
\( \therefore \) The inverse is also symmetric.

1 mark46

If \( A \) is a matrix of order \( 3 \times 3 \), then \( \text{adj}(\text{adj } A) \) is equal to:

1. \( |A| A \)
2. \( |A|^2 A \)
3. \( |A| I \)
4. \( A \)

Solution: Option (A)

\( \because \text{adj}(\text{adj } A) = |A|^{n-2} A \)
For \( n = 3 \):
\( \Rightarrow \text{adj}(\text{adj } A) = |A|^{3-2} A \)
\( \therefore \text{adj}(\text{adj } A) = |A| A \)

1 mark47

If the determinant of a square matrix \( A \) is zero, what can be said about its inverse?

1. It is the null matrix
2. It is the identity matrix
3. It does not exist
4. It is equal to the transpose

Solution: Option (C)

The existence of an inverse requires \( |A| \neq 0 \) (i.e., the matrix must be non-singular).
\( \therefore \) It does not exist.

1 mark48

If \( \text{Trace}(A) = 4 \) and \( \text{Trace}(B) = 5 \), then the value of \( \text{Trace}(3A - 2B) \) is:

1. 22
2. -2
3. 2
4. 12

Solution: Option (C)

\( \because \text{Tr}(kA + mB) = k\text{Tr}(A) + m\text{Tr}(B) \)
\( \Rightarrow \text{Tr}(3A - 2B) = 3(4) - 2(5) \)
\( \Rightarrow \text{Tr}(3A - 2B) = 12 - 10 \)
\( \therefore \text{Tr}(3A - 2B) = 2 \)

1 mark49

The determinant of an identity matrix of any order is always:

1. 0
2. 1
3. Depends on the order
4. Infinity

Solution: Option (B)

An identity matrix is a diagonal matrix where all principal diagonal elements are 1.
\( \Rightarrow |I| = 1 \times 1 \times \dots \times 1 \)
\( \therefore |I| = 1 \)

1 mark50

For any square matrix \( A \), the value of \( (A^{-1})^{-1} \) is:

1. \( I \)
2. \( A^{-1} \)
3. \( A \)
4. \( A^2 \)

Solution: Option (C)

The inverse of the inverse of a matrix returns the original matrix.
\( \therefore (A^{-1})^{-1} = A \)

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