Question

if A and B are symmetric matrices of order n(A not equal to B ) then A+B is zero matrix ,A+B is diagonal matrix, A+B is symmetric, A+B is skew symmetric ​

Answer 1

Answer:

Step-by-step explanation:

Answer is (B) A + B is symmetric A and B are symmetric. A = A′, B = B′. So (A + B)′ = A′ + B′ = A + B

Answer 2

Answer:

A + B is symmetric.

Step-by-step explanation:

Consider the two symmetric matrices $A$ and $B$as follows:

$A=\left[\begin{array}{ccc}a&d&e\\d&b&f\\e&f&c\end{array}\right]$

$B=\left[\begin{array}{ccc}x&p&q\\p&y&r\\q&r&z\end{array}\right]$

Clearly, $A\neq B$.

Compute the sum (addition) of matrices A and B as follows:

$A+B=\left[\begin{array}{ccc}a+x&d+p&e+q\\d+p&b+y&f+r\\e+q&f+r&c+z\end{array}\right]$

(a) $A + B$ is a zero matrix.

Zero matrix: A zero matrix is a matrix all of whose entries are zero.

Observe that all the entries of the matrix A + B are not zero.

Therefore, option (a) is incorrect.

(b) A + B is a diagonal matrix

Diagonal matrix: A matrix in which the non-diagonal entries are all $0$.

Clearly, the non-diagonal entries of the matrix A + B are not zero.

Therefore, option (b) is incorrect.

(c) A + B is symmetric

Symmetric matrix: A matrix is said to $be$ symmetric if $A = A^T$, where $A^T$ is the transpose of of the matrix A.

Since $A$ and $B$ are symmetric matrices. Then,

$A=A^T$ and $B=B^T$ . . . . . (1)

Now,

Consider the transpose of the matrix A+ B as follows:

⇒ $(A+B)^T$

⇒ $A^T+B^T$

From (1), we get

⇒ $A+B$

⇒ $A+B=(A+B)^T$

Thus, A + B is symmetric.

Therefore, option (c) is correct.

(d) A + B is skew symmetric

Skew-symmetric matrix: A matrix is said to be skew-symmetric if $A = -A^T$, where $A^T$ is the transpose of of the matrix A.

Since A + B is symmetric.

So, the matrix A + B cannot be skew-symmetric.

Therefore, option (d) is incorrect.

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