Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Matrices

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

Toolbox:
• A square matrix A=[a$_{ij}$] is said to be symmetric if A'=A that is $[a_{ij}]=[a_{ji}]$ for all possible value of i and j.
• If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A.
Step1:
To prove:
AA' is symmetric matrix
From the property of transpose of matrix we have [$(AB)'=B'A']$
Hence on taking whole transpose of AA' we get
$(AA')'=(A')'A'$
$(A')'A'=AA'$ [we know (A')'=A]
Hence we get (AA')'=AA'
Hence we can say $AA'$ is symmetric,
Step2:
To prove:
A'A is symmetric matrix
From the property of transpose of matrix we have [$(AB)'=B'A']$
Hence on taking whole transpose of AA' we get
$(A'A)'=A'(A')'$
We know that $(A')'=A$
$(A'A)'=A'A$
$\Rightarrow A'A\rightarrow$ is symmetric.
$A'A$ & $AA'$ are symmetric for square matrix.