Question

(a) Show that every square matrix can be expressed as the sum of a symmetric & a skew- symmetric
matrix.
(b) If A & B are two Symmetric matrices of same order, then show that
(i) (AB – BA) is skew-symmetric Matrix.
(ii) (AB + BA) is symmetric Matrix.
plssss helppp someone verryyyy urgent
class 12 matrixx

Answer 1

Step-by-step explanation:

⇒A=A

′

and B=B

′

From the property of transpose of matrices. we have

AB=BA

Now consider AB−BA and by taking transpose of it, we get

$$\left(AB−BA\right)=\left(AB\right)−\left(BA\right)={B}^{\prime}{A}^{\prime}-{A}^{\prime}{B}^{\prime}$$

Replace A

′

→A and B

′

→B

=BA−AB=−(AB−BA) (by taking negative common)

we know that a matrix is said to b skew symmetric matrix if $$A=−A$$

Hence $$AB−BA$$ is skew symmetric matrices