Algebra

Matrices & Determinants

Medium

applied

Which of the following is NOT correct?

If matrix B is the inverse of matrix A, then A is the inverse of B.

A rectangular matrix of order $m \times n$ , $m \neq n$ does not possess an inverse.

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

If a square matrix is invertible, then it may have more than one inverse.

✅ Correct Option: 4

If matrix B is the inverse of matrix A, then AB = I and BA = I (where I is the identity matrix).

This means A also works as the inverse of B.

The statement is correct.

A rectangular matrix of order $m \times n$ where $m \neq n$ cannot have an inverse.

Only square matrices can have inverses. For a matrix to have an inverse, the product must yield an identity matrix, which is always square. A non-square matrix cannot produce a square identity matrix of matching size.

The statement is correct.

For symmetric matrices A and B, where $A^T = A$ and $B^T = B$ :

Let $C = AB - BA$

Taking transpose:

$C^T = (AB - BA)^T$

$C^T = (AB)^T - (BA)^T$

$C^T = B^T A^T - A^T B^T$

Since A and B are symmetric:

$C^T = BA - AB$

$C^T = -(AB - BA)$

$C^T = -C$

Therefore $(AB - BA)$ is skew-symmetric.

The statement is correct.

An invertible matrix has exactly one unique inverse, not more than one.

Suppose matrix A has two different inverses B and C, where $AB = I$ and $AC = I$ .

Multiplying $AB = I$ on the left by C:

$C(AB) = CI$

$(CA)B = C$

$IB = C$

$B = C$

This shows B and C must be identical. An invertible matrix cannot have more than one inverse.

The statement is not correct.

The answer is option 4.

2026: 21 May Shift 1

If $\begin{vmatrix} 2 & 3 & 2 \\ x & x & x \\ 4 & 9 & 1 \end{vmatrix} + \begin{vmatrix} 3 & 2 \\ 4 & 5 \end{vmatrix} = 0$ , then the value of $x$ is

2025: 13 May Shift 1

If $A = [a_{ij}]_{3×3}$ where $a_{ij} = \begin{cases} (-1)^{i+j} - 1, & i = j \\ (-1)^{i+j}, & i \neq j \end{cases}$ then the value of $A + A^T$ is:

2025: 27 May Shift 1

If $c_{ij}$ denotes the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 0 & -3 & 2 \\ 4 & 2 & 3 \end{bmatrix}$ then the value of $c_{21} \cdot c_{33}$ is

2026: 26 May Shift 2

Which of the following statements are correct ?

A. If the product of 2 non-zero square matrices is a zero matrix, then either A is singular or B is singular

B. If A is a non singular matrix , then $|A^{-1}| = \dfrac{1}{|A|}$

C. If A is an invertible square matrix then $adj\,A^T = \dfrac{1}{adj\,A}$

D. If A and B are invertible matrices then $(AB)^{-1} = B^{-1}A^{-1}$

Choose the correct answer from the options given below:

2025: 22 May Shift 1

Let $A = \begin{bmatrix} 152 & 105 & 3 \\ 149 & 25 & 35 \\ 2 & 1 & 0 \end{bmatrix}$ . If $A_{ij}$ denotes the co-factor of an element $a_{ij}$ of the matrix A, then the value of $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}$ is equal to

2025: 14 May Shift 1

If A and B are skew-symmetric matrices, then which one of the following is NOT true?

Which of the following is NOT correct?

Solution

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