Algebra

Matrices & Determinants

Medium

applied

If A and B are symmetric matrices of the same order, then which one of the following is true?

A+B is a skew-symmetric matrix

A-B is skew-symmetric matrix.

AB-BA is a symmetric matrix.

AB+BA is a symmetric matrix.

✅ Correct Option: 4

A symmetric matrix satisfies $A^T = A$ (the matrix equals its transpose).

A skew-symmetric matrix satisfies $A^T = -A$ (transpose gives the negative).

Key properties:

$(A + B)^T = A^T + B^T$

$(AB)^T = B^T A^T$ (order reverses)

Given that $A$ and $B$ are both symmetric matrices:

$A^T = A$

$B^T = B$

Testing $A+B$ :

$(A+B)^T = A^T + B^T$

$= A + B$

Since $(A+B)^T = (A+B)$ , this is symmetric, not skew-symmetric.

Testing $A-B$ :

$(A-B)^T = A^T - B^T$

$= A - B$

Since $(A-B)^T = (A-B)$ , this is symmetric, not skew-symmetric.

Testing $AB-BA$ :

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

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

$= BA - AB$

$= -(AB - BA)$

Since $(AB-BA)^T = -(AB-BA)$ , this is skew-symmetric, not symmetric.

Testing $AB+BA$ :

$(AB+BA)^T = (AB)^T + (BA)^T$

$= B^T A^T + A^T B^T$

$= BA + AB$

$= AB + BA$

Since $(AB+BA)^T = (AB+BA)$ , this is symmetric.

Therefore, $AB+BA$ is a symmetric matrix.

2025: 21 May Shift 1

If A and B are symmetric matrices of same order, then which of the following are correct?

(A) AB-BA is a skew-symmetric matrix.

(B) AB+BA is a skew-symmetric matrix.

(D) AB+BA is a symmetric matrix.

Choose the correct answer from the options given below:

2026: 18 May Shift 2

If $\begin{vmatrix} m! & (m+1)! & (m+2)! \\ (m+1)! & (m+2)! & (m+3)! \\ (m+2)! & (m+3)! & (m+4)! \end{vmatrix} = 2k$ , where $m$ is a positive integer then which of the following are factors of $k$ ?

A. $m!$

B. $(m+1)!$

C. $(m+2)!$

D. $(m+3)!$

Choose the correct answer from the options given below:

2025: 15 May Shift 1

The value of $\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$ is

2025: 14 May Shift 2

If $A = \begin{bmatrix} 2 & -3 & 4 \\ -3 & 5 & x \\ 4 & 3 & 0 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 2 & -10 \\ -2 & z & 6 \\ y & -6 & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the value of $(xy + yz + zx)$ is

2022: 6 Aug Shift 2

The set of all values of $\alpha$ for which the system of linear equations

$x + y + z = 1$

$x + 2y + 4z = \alpha$

$x + 4y + 10z = \alpha^2$

is consistent, is

2023: 15 June Shift 2

The value of the determinant $\begin{vmatrix} a\cos\theta & b\sin\theta & 0 \\ -b\sin\theta & a\cos\theta & 0 \\ 0 & 0 & c \end{vmatrix}$ is :

If A and B are symmetric matrices of the same order, then which one of the following is true?

Solution

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