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If $A$ is a skew-symmetric matrix of order $n$, and $C$ is a column matrix of order $n \times 1$, then $C^T A C$ is
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Verified Answer
The correct answer is:
A zero matrix of order one
Since A is a Skew-Symmetric matrix
$\mathrm{A}^{\mathrm{T}}=-\mathrm{A}$
$\mathrm{C}$ is a column matrix of order $\mathrm{n} \times 1$
So order of $C^{\mathrm{T}}=1 \times n$
So order of ${C}^T A C=(1 \times n) \times(n \times n) \times(n \times 1)$ $=1 \times 1$
Since $\mathrm{C}^{\mathrm{T}} \mathrm{AC}$ has order $=1 \times 1$
$\therefore\left(\mathrm{C}^{\mathrm{T}} \mathrm{AC}\right)^{\mathrm{T}}=\mathrm{C}^{\mathrm{T}} \mathrm{AC}$
Let's assume $\mathrm{C}^{\mathrm{T}} \mathrm{AC}=\mathrm{B}$
$\left(C^T A C\right)^T=B^T$
or $C^T A^T\left(C^T\right)^T=B^T$
or $\mathrm{C}^{\mathrm{T}} \mathrm{A}^{\mathrm{T}} \mathrm{C}=\mathrm{B}^{\mathrm{T}}$
or $\mathrm{C}^{\mathrm{T}}(-\mathrm{A}) \mathrm{C}=\mathrm{B}^{\mathrm{T}}\left[\because \mathrm{A}^{\mathrm{T}}=-\mathrm{A}\right]$
or $-\mathrm{C}^{\mathrm{T}} \mathrm{AC}=\mathrm{B}\left[\because \mathrm{B}^{\mathrm{T}}=\mathrm{B}\right]$
or $-B=B$
or $2 B=0$
or $B=0$
(3) a zero matrix of order 1
$\mathrm{A}^{\mathrm{T}}=-\mathrm{A}$
$\mathrm{C}$ is a column matrix of order $\mathrm{n} \times 1$
So order of $C^{\mathrm{T}}=1 \times n$
So order of ${C}^T A C=(1 \times n) \times(n \times n) \times(n \times 1)$ $=1 \times 1$
Since $\mathrm{C}^{\mathrm{T}} \mathrm{AC}$ has order $=1 \times 1$
$\therefore\left(\mathrm{C}^{\mathrm{T}} \mathrm{AC}\right)^{\mathrm{T}}=\mathrm{C}^{\mathrm{T}} \mathrm{AC}$
Let's assume $\mathrm{C}^{\mathrm{T}} \mathrm{AC}=\mathrm{B}$
$\left(C^T A C\right)^T=B^T$
or $C^T A^T\left(C^T\right)^T=B^T$
or $\mathrm{C}^{\mathrm{T}} \mathrm{A}^{\mathrm{T}} \mathrm{C}=\mathrm{B}^{\mathrm{T}}$
or $\mathrm{C}^{\mathrm{T}}(-\mathrm{A}) \mathrm{C}=\mathrm{B}^{\mathrm{T}}\left[\because \mathrm{A}^{\mathrm{T}}=-\mathrm{A}\right]$
or $-\mathrm{C}^{\mathrm{T}} \mathrm{AC}=\mathrm{B}\left[\because \mathrm{B}^{\mathrm{T}}=\mathrm{B}\right]$
or $-B=B$
or $2 B=0$
or $B=0$
(3) a zero matrix of order 1
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