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Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

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$A^{2} = I$

∴ A^{3} = A^{2} A = I A = A

We know

(A + B)^{3} = A^{3} + 3 A^{2} B + 3 A B^{2} + B^{3}

(A - B)^{3} = A^{3} - 3 A^{2} B + 3 A B^{2} - B^{2}

Provided that

A B = B A

Since,

A I = I A = A

∴ (A + I)^{3} = A^{3} + 3 A^{2} I + 3 A I^{2} + I^{3}

and,

(A - I)^{3} = A^{3} - 3 A^{2} I + 3 A I^{2} - I^{3}

\Rightarrow (A + I)^{3} = A^{3} + 3 A^{2} + 3 A + I

and ,

(A - I)^{3} = A^{3} - 3 A^{2} + 3 A - I

∴ (A + I)^{3} + (A - I)^{3} = 2 (A^{3} + 3 A)

= 2 (A + 3 A)

Hence,

= 8 A

(A - I)^{3} + (A + I)^{3} - 7 A = 8 A - 7 A = A

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