Question

Let $A=\left[\begin{array}{ccc}X& Y& Z\end{array}\right]$ be a $3\times 3$ orthogonal matrix with $X,Y,Z$ as its column vectors. Then $B=X{X}^T+Y{Y}^T$

• A. is a symmetric matrix
• B. is the $3\times 3$ identity matrix
• C. satisfies $B^2=B$
• D. satisfies $BZ=O,O$ being the null matrix

Answer 1

Answer: (A), (C), (D)

satisfies $BZ=O,O$ being the null matrix

$A$ is orthogonal matrix.
$\Rightarrow A{A}^T=A^{T}A=I$
$\Rightarrow X^{T}X=Y^{T}Y=Z^{T}Z=1$
and $X^{T}Y=X^{T}Z=Y^{T}Z=Y^{T}X=Z^{T}X=Z^{T}Y=0$

$B=X{X}^T+Y{Y}^T$
$B^T=(X{X}^T+Y{Y}^T{)}^T=X{X}^T+Y{Y}^T$
Hence, $B$ is symmetric.

$B^2=(X{X}^T+Y{Y}^T)(X{X}^T+Y{Y}^T)$
$=X{X}^T(X{X}^T+Y{Y}^T)+Y{Y}^T(X{X}^T+Y{Y}^T)$
$=X{X}^T+Y{Y}^T=B$

$BZ=(X{X}^T+Y{Y}^T)Z$
$=X{X}^{T}Z+Y{Y}^{T}Z$
$=O$

We know that for any $n\times n$ identity matrix $I$ and a $n\times 1$ non-zero vector $A$, $IA=A$
But here, $BZ=O$
Hence, $B$ can't be the $3\times 3$ identity matrix.