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

Engineering

Mathematics

Algebra of Matrices

Special Type of Square 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.