Algebra

Matrices & Determinants

Medium

applied

If A and B are symmetric matrices of the same order, then

(AB+BA) is skew symmetric matrix.

(AB+BA) is always a diagonal matrix.

(AB+BA) is a symmetric matrix.

(AB-BA) is a symmetric matrix.

✅ Correct Option: 3

A matrix is symmetric when it equals its own transpose. For symmetric matrices $A$ and $B$ :

$A^T = A$ and $B^T = B$

To check if $(AB+BA)$ is symmetric, take the transpose:

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

Using the transpose property $(XY)^T = Y^T X^T$ :

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

Since $A^T = A$ and $B^T = B$ :

$B^T A^T + A^T B^T = BA + AB$

$= AB + BA$

Therefore $(AB+BA)^T = (AB+BA)$ , which means $(AB+BA)$ is symmetric.

For $(AB-BA)$ :

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

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

$= BA - AB$

$= -(AB - BA)$

This shows $(AB-BA)$ is skew-symmetric, not symmetric.

$(AB+BA)$ is not necessarily diagonal since this requires $AB = BA$ , which is not always true for symmetric matrices.

Therefore, $(AB+BA)$ is symmetric.

2026: 22 May Shift 2

If $f(x) = \begin{vmatrix} 0 & x+a & x+b \\ x-a & 0 & x+c \\ x-b & x-c & 0 \end{vmatrix}$ then the value of $f(0)$ is equal to:

2026: 18 May Shift 1

If $A$ is a square matrix of order 3 such that $A(adj\ A) = \begin{bmatrix}-2 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & -2\end{bmatrix}$ , then $|adj\ A|$ is equal to

2025: 21 May Shift 2

If $A$ is a $3 \times 3$ matrix such that $|adj A| = 9$ and $|kA^{-1}| = 9$ , then the value of $k$ are:

2025: 19 May Shift 1

Which of the following statements are correct?

(A) If A is a square matrix, then $|A^2| = |A|^2$ .

(B) If A and B are square matrices of the same order, then det (AB) = det (A) + det (B).

(D) If the matrix $\begin{bmatrix} 5 -x & x -1 \\ 3 &5 \end{bmatrix}$ is singular, then the value of x is 7/2.

Choose the correct answer from the options given below:

2026: 21 May Shift 2

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2025: 22 May Shift 2

If A and B are symmetric matrices of order 3 x 3 then the matrix $2AB - BA$ is:

If A and B are symmetric matrices of the same order, then

Solution

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