I Show that the matrix- A-beginbmatrix 18-1 - 511-1 - 2 - 1 l5 - 1 - 3 end bmatrix is a symmetric matrix0 - 1 -111-1 - 0 - 1111 -1 - 0endbmatrix is a skew -symmetric matrix-

(i) Here, A'=[ 1 amp; 1 amp; 5; 1 amp; 2 amp; 1; 5 amp; 1 amp; 3 ]=[ 1 amp; 1 amp; 5; 1 amp; 2 amp; 1; 5 amp; 1 ...

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Standard XII

Mathematics

Skew Symmetric Matrix

i Show that t...

Question

(i) Show that the matrix, $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 5 - 1 & 2 & 1 5 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$ is a symmetric matrix

(ii) Show that the matrix, $A = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & - 1 - 1 & 0 & 1 1 & - 1 & 0 \end{matrix} ⎤ ⎥ ⎦$ is a skew -symmetric matrix.

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Solution

(i) Here, $A^{'} = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 5 - 1 & 2 & 1 5 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 5 - 1 & 2 & 1 5 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = a ∴ A^{'} = A$
Hence, A is a symmetric matrix.

(ii) Now, $A^{'} = {⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & - 1 - 1 & 0 & 1 1 & - 1 & 0 \end{matrix} ⎤ ⎥ ⎦}^{'} = ⎡ ⎢ ⎣ \begin{matrix} 0 & - 1 & 1 1 & 0 & - 1 - 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦ = - ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & - 1 - 1 & 0 & 1 1 & - 1 & 0 \end{matrix} ⎤ ⎥ ⎦ = - A$
$∴ A^{'} = - A$ . Hence, A is skew-symmetric matrix.

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Similar questions

Show that the matrix, $A = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & - 1 - 1 & 0 & 1 1 & - 1 & 0 \end{matrix} ⎤ ⎥ ⎦$ is a skew -symmetric matrix.

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

$⎡ ⎢ ⎣ \begin{matrix} 3 & 3 & - 1 - 2 & - 2 & 1 - 4 & - 5 & 2 \end{matrix} ⎤ ⎥ ⎦$

Q. (i) Show that the matrix

A = ⎡ ⎢ ⎣ \begin{matrix} 1 - 1 5 \end{matrix} \begin{matrix} - 1 21 \end{matrix} \begin{matrix} 513 \end{matrix} ⎤ ⎥ ⎦

is a symmetric matrix

(ii) Show that the matrix

A = ⎡ ⎢ ⎣ \begin{matrix} 0 - 1 1 \end{matrix} \begin{matrix} 10 - 1 \end{matrix} \begin{matrix} - 1 10 \end{matrix} ⎤ ⎥ ⎦

is a skew symmetric matrix.

(i) Show that the matrix is a symmetric matrix

(ii) Show that the matrix is a skew symmetric matrix

If the matrix A = $⎡ ⎢ ⎣ \begin{matrix} 0 & a & - 3 2 & 0 & - 1 b & 1 & 0 \end{matrix} ⎤ ⎥ ⎦$ is skew symmetric, find the value of 'a' and 'b'.

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(i) Show that the matrix, A=⎡⎢⎣1−15−121513⎤⎥⎦ is a symmetric matrix (ii) Show that the matrix, A=⎡⎢⎣01−1−1011−10⎤⎥⎦ is a skew -symmetric matrix.

(i) Show that the matrix, $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 5 - 1 & 2 & 1 5 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$ is a symmetric matrix

(ii) Show that the matrix, $A = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & - 1 - 1 & 0 & 1 1 & - 1 & 0 \end{matrix} ⎤ ⎥ ⎦$ is a skew -symmetric matrix.