If A = 1 2 0 0 1 3 − 2 5 3 , then verify that A + A ' is symmetric.

Given A = 1 2 0 0 1 3 − 2 5 3 A matrix is said to be symmetric, if C = C ' To prove that A + A ' = A + A ' ' A ' = 1 0 - 2 2 1 5 0 3 3 A + A ' = 1 2 0 0 1 3 - 2 5 3 + 1 0 - 2 2 1 5 0 3 3 = 2 2 - 2 2 2 8 - 2 8 6 A + A ' ' = 2 2 - 2 2 2 8 - 2 8 6 ∴ A + A ' = A + A ' ' Hence, proved.

If A = 1 2 0 0 1 3 − 2 5 3 , then verify that A - A ' is skew-symmetric.

Given A = 1 2 0 0 1 3 − 2 5 3 A matrix is said to be skew-symmetric, if C ' = - C To prove that A - A ' ' = - A - A ' A ' = 1 0 - 2 2 1 5 0 3 3 A - A ' = 1 2 0 0 1 3 - 2 5 3 - 1 0 - 2 2 1 5 0 3 3 = 0 2 2 - 2 0 - 2 - 2 2 0 A - A ' ' = 0 - 2 - 2 2 0 2 2 - 2 0 = - 0 2 2 - 2 0 - 2 - 2 2 0 = - A - A ' ∴ A - A ' ' = - A - A ' Hence, proved.

E M B I B E

Elements of Mathematics Class 12>Chapter 4 - Matrices>Exercises - 4 (b)>Q 2

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IMPORTANT

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If $A$ is symmetric then $A - A^{'}$ is both symmetric and skew symmetric.

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Important Questions on Matrices

EASY

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IMPORTANT

Elements of Mathematics Class 12>Chapter 4 - Matrices>Exercises - 4 (b)>Q 2

For any square matrix

{(A - A^{'})}^{2}

is skew symmetric.

EASY

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IMPORTANT

Elements of Mathematics Class 12>Chapter 4 - Matrices>Exercises - 4 (b)>Q 2

A matrix which is not symmetric is skew symmetric.

MEDIUM

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IMPORTANT

Elements of Mathematics Class 12>Chapter 4 - Matrices>Exercises - 4 (b)>Q 3

If a symmetric/skew symmetric matrix is expressed as a sum of symmetric and skew symmetric matrix then prove that one of the matrices in the sum must be zero.

EASY

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IMPORTANT

Elements of Mathematics Class 12>Chapter 4 - Matrices>Exercises - 4 (b)>Q 4

$A$ and $B$ are square matrices of same order, prove that:

If $A, B and A B$ are all symmetric, then $A B - B A = 0$ .