Let $A=\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]$. Express $A$ as the sum of two matrices such that one is symmetric and the other is skew symmetric.

$P=\left[\begin{array}{ccc}3 & 3 & 5/2 \\ 3 & 1 & 9/2 \\ 5/2 & 9/2 & 7\end{array}\right],\ Q=\left[\begin{array}{ccc}0 & -1 & 5/2 \\ 1 & 0 & -3/2 \\ -5/2 & 3/2 & 0\end{array}\right]$

CBSE Class 12 · 2008 · MatricesLong Answer

WhatsApp Email

Let

A = 340216537

. Express

A

as the sum of two matrices such that one is symmetric and the other is skew symmetric.

Solution

Write

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

with

A^{'} = 325413067

P = \frac{1}{2} (A + A^{'}) = 33 \frac{5}{2} 31 \frac{9}{2} \frac{5}{2} \frac{9}{2} 7 (symmetric), Q = \frac{1}{2} (A - A^{'}) = 01 - \frac{5}{2} - 1 0 \frac{3}{2} \frac{5}{2} - \frac{3}{2} 0 (skew) .

Then

A = P + Q

CBSE Class 12 Board · 2008 · Subjective

Practise more: all CBSE board PYQs on Matrices →

More board questions on Matrices

The product of matrix $P$ and $Q$ is equal to a diagonal matrix. If the order of matrix $Q$ is $3 \times 2$ , then order of matrix $P$ is:
CBSE 2024View solution →
If inverse of matrix $7 - 1 - 1 - 3 10 - 3 01$ is the matrix $111 3 λ 3 334$ , then value of $λ$ is:
CBSE 2024View solution →
Find the matrix $A^{2}$ , where $A = [a_{ij}]$ is a $2 \times 2$ matrix whose elements are given by $a_{ij} = max (i, j) - min (i, j)$ :
CBSE 2024View solution →
If $A$ is a square matrix of order $3$ such that the value of $∣ adj A ∣ = 8$ , then the value of $∣ A^{⊤} ∣$ is:
CBSE 2024View solution →
If $A = 123 23 - 3 - 3 2 - 4$ , then find $A^{- 1}$ and hence solve the following system of equations:
$x + 2 y - 3 z = - 4, 2 x + 3 y + 2 z = 2, 3 x - 3 y - 4 z = 11.$
CBSE 2024View solution →
If $A$ is a $2 \times 3$ matrix such that $A B$ and $A B^{'}$ both are defined, then order of matrix $B$ is:
CBSE 2023View solution →