Name: Adding and Subtracting Matrices
Uploaded: 2019-07-19
Duration: 6 min 12 s
Description: Two matrices with the same order can be added or subtracted. Watch this video to learn some examples based on the addition and subtraction of matrices. Let's...

Question

Express the following matrix as the sum of a symmetric and a skew-symmetric matrix $[\begin{array}{r} 1 & 3 & 5 \\ - 6 & 8 & 3 \\ - 4 & 6 & 5 \end{array}]$

Accepted Answer

Let $A$ be any square matrix, then sum of symmetric and skew-symmetric matrix

$A = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})$ .

Let $A = (\begin{array}{lll} 1 & 3 & 5 \\ - 6 & 8 & 3 \\ - 4 & 6 & 5 \end{array})$

Then, $A^{'} = (\begin{array}{l} 1 & - 6 & - 4 \\ 3 & 8 & 6 \\ 5 & 3 & 5 \end{array})$

Now, $\frac{1}{2} [A + A^{'}] = \frac{1}{2} [(\begin{array}{lll} 1 & 3 & 5 \\ - 6 & 8 & 3 \\ - 4 & 6 & 5 \end{array}) + (\begin{array}{l} 1 & - 6 & - 4 \\ 3 & 8 & 6 \\ 5 & 3 & 5 \end{array})] = \frac{1}{2} (\begin{matrix} 2 & - 3 & 1 \\ - 3 & 16 & 9 \\ 1 & 9 & 10 \end{matrix}) = (\begin{array}{ccc} 1 & - \frac{3}{2} & \frac{1}{2} \\ - \frac{3}{2} & 8 & \frac{9}{2} \\ \frac{1}{2} & \frac{9}{2} & 5 \end{array})$

And $\frac{1}{2} [A - A^{'}] = \frac{1}{2} [(\begin{array}{lll} 1 & 3 & 5 \\ - 6 & 8 & 3 \\ - 4 & 6 & 5 \end{array}) - (\begin{array}{l} 1 & - 6 & - 4 \\ 3 & 8 & 6 \\ 5 & 3 & 5 \end{array})] = \frac{1}{2} (\begin{matrix} 0 & 9 & 9 \\ - 9 & 0 & - 3 \\ - 9 & 3 & 0 \end{matrix}) = (\begin{array}{ccc} 0 & \frac{9}{2} & \frac{9}{2} \\ - \frac{9}{2} & 0 & - \frac{3}{2} \\ - \frac{9}{2} & \frac{3}{2} & 0 \end{array})$

$\Rightarrow A = (\begin{array}{ccc} 1 & - \frac{3}{2} & \frac{1}{2} \\ - \frac{3}{2} & 8 & \frac{9}{2} \\ \frac{1}{2} & \frac{9}{2} & 5 \end{array}) + (\begin{array}{ccc} 0 & \frac{9}{2} & \frac{9}{2} \\ - \frac{9}{2} & 0 & - \frac{3}{2} \\ - \frac{9}{2} & \frac{3}{2} & 0 \end{array})$ .

Express the following matrix as the sum of a symmetric and a skew-symmetric matrix $[\begin{array}{r} 1 & 3 & 5 \\ - 6 & 8 & 3 \\ - 4 & 6 & 5 \end{array}]$

Important Questions on Elementary Theory of Matrices

Learn and Prepare for any exam you want !

Express the following matrix as the sum of a symmetric and a skew-symmetric matrix 135-683-465

Important Questions on Elementary Theory of Matrices

Learn and Prepare for any exam you want !

Express the following matrix as the sum of a symmetric and a skew-symmetric matrix $[\begin{array}{r} 1 & 3 & 5 \\ - 6 & 8 & 3 \\ - 4 & 6 & 5 \end{array}]$