Download App

Algebra of Matrices — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices3 Marks
Question
Define a symmetric matrix. Prove that for $\text{A}=\begin{bmatrix}2&4 \\5&6 \end{bmatrix},$ $A + A^T$ is a symmetric matrix where $A^T​​​​​​​$​​​​​​​ is the transpose of $A$.
✓

Answer

A squrae matrix A is called a syammetric matrix, if $A^T = A$.
Given: $\text{A}=\begin{bmatrix}2&4 \\5&6 \end{bmatrix}$
$\text{A}^{\text{T}}=\begin{bmatrix}2&5 \\4&6 \end{bmatrix}$
Now, $\text{A}+\text{A}^{\text{T}}=\begin{bmatrix}2&4 \\5&6 \end{bmatrix}+\begin{bmatrix}2&5 \\4&6 \end{bmatrix}$
$\Rightarrow\text{A}+\text{A}^{\text{T}}=\begin{bmatrix}4 & 9 \\9&12 \end{bmatrix}\ \dots(\text{i})$ $(\text{A}+\text{A}^{\text{T}})^{\text{T}}=\begin{bmatrix}4 & 9 \\9&12 \end{bmatrix}^{\text{T}}$ $=\begin{bmatrix}4 & 9 \\9&12 \end{bmatrix}$
$\Rightarrow\text{A}+\text{A}^{\text{T}} $ [From eq. (1)]
$\therefore\ (\text{A}+\text{A}^{\text{T}})^{\text{T}}=(\text{A}+\text{A}^{\text{T}})$
Thus, $(A + A^T)$ is a symmetric matrix.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(3 Marks)" from Algebra of Matrices→Algebra of Matrices — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evalute the following integrals:
$\int\frac{\sin2\text{x}}{\sin\Big(\text{x}-\frac{\pi}{6}\Big)\sin\Big(\text{x}+\frac{\pi}{6}\Big)}\text{dx}$
→
Find the mean and standard deviation of the following probability distributions:
$x_i$ -2 -1 0 1 2
$p_i$ 0.1 0.2 0.4 0.2 0.1
→
Find the area of the region bounded by the parabola:

$y=4-x^2$ and the $X$-axis.

→
Solve the following differential equation
$\text{x}\frac{\text{dy}}{\text{dx}}+\cot\text{y}=0$
→
The cartesian equations of a line are x = ay + b, z = cy + d. Find its direction ratios and reduce it to vector form.
→
Show that the points $(1,-1,3)$ and $(3,4,3)$ are equidistant from the plane $5 x+2 y-7 z+8=0$.
→
Construct the switching circuit of the following : p ∨ (~p ) ∨ (~q) ∨ (p ∧ q)
→
Diffrentiate the following w. r. t. x.

$\tan ^{-1}\left(\frac{2 \sqrt{x}}{1+3 x}\right)$

→
If $\vec{\text{c}}$ is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{b}},$ then prove that it is perpendicular to both $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}.$
→
Find the principal solutions of the following equations :

$\sin \theta=-\frac{1}{2}$

→