Show that A'A and AA' are both symmetric matrices for any matrix A.

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

Show that A'A and AA' are both symmetric matrices for any matrix A.

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Let $A = [\begin{array}{ll} a b \\ c d \end{array}], A^{'} = [\begin{array}{ll} a c \\ b d \end{array}]$

$A^{'} \cdot A = [\begin{array}{ll} a c \\ b d \end{array}] \cdot [\begin{array}{ll} a b \\ c d \end{array}] = [\begin{array}{ll} a^{2} + c^{2} ab + cd \\ ab + cd b^{2} + d^{2} \end{array}]$

and

${AA}^{'} = [\begin{array}{ll} a b \\ c d \end{array}] \cdot [\begin{array}{ll} a c \\ b d \end{array}] = [\begin{array}{ll} a^{2} + b^{2} ac + bd \\ ac + bd c^{2} + d^{2} \end{array}]$

for symmetric matrices,

(B)' = B

So, ${(A^{'} A)}^{'} = [\begin{array}{ll} a^{2} + c^{2} ab + cd \\ ab + cd b^{2} + d^{2} \end{array}] = A^{'} A$ (symmetric matrix)

and

${({AA}^{'})}^{'} = [\begin{array}{cc} a^{2} + b^{2} & ac + bd \\ ac + bd & c^{2} + d^{2} \end{array}] = A \cdot A^{'}$ (symmetric matrix)

Hence, proved that AA' and AA' both are symmetric matrices.

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