If A is symmetric matrix, then B'AB is _____.

NCERT Exemplar · Q75

A

Telangana TsbieShort· 1mImportance★★★★★

Appeared in past exams:CBSE 2024· Set 65/2/1· 1mexact

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Transposing $B'AB$ gives back $B'AB$ (because $A'=A$ ), so $B'AB$ is symmetric.

What to check

A matrix $M$ is symmetric when $M'=M$ . So set $M=B'AB$ and compute its transpose.

Apply the reversal law

The transpose of a product reverses the factor order: $(XYZ)'=Z'Y'X'$ . With $X=B'$ , $Y=A$ , $Z=B$ ,

(B'AB)'=B'\,A'\,(B')'.

Now simplify each piece:

$(B')'=B$ (transposing twice returns the original), so the last factor becomes $B$ ;
$A'=A$ , because $A$ is symmetric (given).

Hence

(B'AB)'=B'\,A\,B=B'AB.

Conclusion …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

TG EAPCET 2022Set eng-2022-07-19-FN1 markMCQ
$\begin{bmatrix} -1 & 2 & b \\ a & 5 & 6 \\ 3 & c & 7 \end{bmatrix}$
›Reveal solutionSolution
A symmetric matrix equals its transpose, which forces $a = 2$ , $b = 3$ , $c = 6$ . Substituting these into the given $3 \times 3$ determinant and evaluating gives the value $-143$ .

The key idea is that symmetry is a condition on the entries: for a matrix $A$ , $A = A^T$ means $A_{ij} = A_{ji}$ for every pair of indices. Once we extract the values of $a$ , $b$ , and $c$ from that condition, the problem reduces to a straightforward determinant calculation.

Let the given matrix be
$M = \begin{bmatrix} -1 & 2 & b \\ a & 5 & 6 \\ 3 & c & 7 \end{bmatrix}.$
For $M$ to be symmetric, we must have $M_{ij} = M_{ji}$ for all $i, j$ .

Compare the $(1,2)$ and $(2,1)$ entries. $M_{12} = 2$ and $M_{21} = a$ . Symmetry demands $2 = a$ , so

$a = 2.$

Compare the $(1,3)$ and $(3,1)$ entries. $M_{13} = b$ and $M_{31} = 3$ . Hence

$b = 3.$

Compare the $(2,3)$ and $(3,2)$ entries. $M_{23} = 6$ and $M_{32} = c$ . Thus

$c = 6.$
All other positions are already symmetric by construction (the diagonal entries are free, and the remaining off-diagonal pairs match automatically once these three equalities hold).

Now substitute $a = 2$ , $b = 3$ , $c = 6$ into the determinant we need:
$\Delta = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = \begin{vmatrix} 2 & 3 & 6 \\ 3 & 6 & 2 \\ 6 & 2 & 3 \end{vmatrix}.$

Evaluate the determinant. A clean way is to use the formula for a $3 \times 3$ determinant directly: …
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TG EAPCET 2023Set eng-2023-05-13-FN1 markMCQ
$A = \begin{bmatrix} 1 & 2 & -1 \\ -1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix}$
›Reveal solutionSolution

Computing $A^2$ directly gives exactly the matrix $A+B$ , so $A^2=A+B$ .

With $A=\begin{bmatrix}1&2&-1\\-1&0&2\\1&2&0\end{bmatrix}$ , multiply $A\cdot A$ entry by entry:
$A^2=\begin{bmatrix}-2&0&3\\1&2&1\\-1&2&3\end{bmatrix}.$
For example, the $(1,1)$ entry is $1(1)+2(-1)+(-1)(1)=-2$ , and the $(2,2)$ entry is $-1(2)+0(0)+2(2)=2$ .

Now add $A$ and $B$ : …
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Q.If $A$ is symmetric matrix, then $B'AB$ is _________.

Concept understanding — Symmetric And Skew Symmetric Matrices

Symmetric and Skew-Symmetric Matrices

Symmetric matrix

Skew-symmetric matrix

Key facts

What to check

Apply the reversal law

Conclusion …

Method: Test the symmetry of a triple product $B'AB$

Steps

Common Mistakes

Q.If AAA is symmetric matrix, then B′ABB'ABB′AB is _________.

What to check

Apply the reversal law

Conclusion …

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Q.If $A$ is symmetric matrix, then $B'AB$ is _________.

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