Name: Elementary Operations or Transformations of a Matrix
Uploaded: 2019-07-19
Duration: 12 min
Description: Learn all about the elementary matrix operations. This becomes quite useful to solve systems of linear equations, or to find the inverse of a matrix.

Question

Express the matrix

A = [\begin{matrix} 2 & 3 \\ - 1 & 4 \end{matrix}]

as a sum of a symmetric and a skew-symmetric matrix.

Accepted Answer

Given matrix is $A = [\begin{matrix} 2 & 3 \\ - 1 & 4 \end{matrix}]$

We need to express this matrix as a sum of a symmetric and a skew-symmetric matrix.

We know that the transpose of matrix $A$ is denoted by $A^{'}$ , and is obtained by interchanging the rows and columns of the matrix $A$ .

Thus,

$A^{'} = [\begin{matrix} 2 & - 1 \\ 3 & 4 \end{matrix}]$

$A + A^{'} = [\begin{matrix} 2 & 3 \\ - 1 & 4 \end{matrix}] + [\begin{matrix} 2 & - 1 \\ 3 & 4 \end{matrix}]$

$= [\begin{matrix} 4 & 2 \\ 2 & 8 \end{matrix}]$

$\frac{1}{2} (A + A^{'}) = \frac{1}{2} [\begin{matrix} 4 & 2 \\ 2 & 8 \end{matrix}]$

$= [\begin{matrix} 2 & 1 \\ 1 & 4 \end{matrix}]$

Let $P = [\begin{matrix} 2 & 1 \\ 1 & 4 \end{matrix}]$

We also observe that $P = P^{'}$

$∴$ $P$ is a symmetric matrix.

Now,

$A - A^{'} = [\begin{matrix} 2 & 3 \\ - 1 & 4 \end{matrix}] - [\begin{matrix} 2 & - 1 \\ 3 & 4 \end{matrix}]$

$= [\begin{matrix} 0 & 4 \\ - 4 & 0 \end{matrix}]$

$\frac{1}{2} (A - A^{'}) = \frac{1}{2} [\begin{matrix} 0 & 4 \\ - 4 & 0 \end{matrix}] = [\begin{matrix} 0 & 2 \\ - 2 & 0 \end{matrix}]$

Let $Q = [\begin{matrix} 0 & 2 \\ - 2 & 0 \end{matrix}]$

Now, $Q^{'} = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}] = - [\begin{matrix} 0 & 2 \\ - 2 & 0 \end{matrix}] = - Q$

$\Rightarrow Q$ is a skew-symmetric matrix.

Hence, $A = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'}) = P + Q$ , where $P$ is a symmetric matrix, and $Q$ is a skew-symmetric matrix.

i.e., $[\begin{matrix} 2 & 3 \\ - 1 & 4 \end{matrix}] = [\begin{matrix} 2 & 1 \\ 1 & 4 \end{matrix}] + [\begin{matrix} 0 & 2 \\ - 2 & 0 \end{matrix}]$

Express the matrix $A = [\begin{matrix} 2 & 3 \\ - 1 & 4 \end{matrix}]$ as a sum of a symmetric and a skew-symmetric matrix.

Important Questions on Matrices

Learn and Prepare for any exam you want !

Express the matrix A=23-14 as a sum of a symmetric and a skew-symmetric matrix.

Important Questions on Matrices

Learn and Prepare for any exam you want !

Express the matrix $A = [\begin{matrix} 2 & 3 \\ - 1 & 4 \end{matrix}]$ as a sum of a symmetric and a skew-symmetric matrix.