Download App

MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
Let $\text{A}=\begin{bmatrix}1&-1&0\\2&1&3\\1&2&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&2&3\\2&1&3\\0&1&1\end{bmatrix},$ Find $A^T, B^T$​​​​​​​ and verify that.$(\text{A}\text{B})^\text{T}=\text{B}^\text{T}+\text{A}^\text{T}$
✓

Answer

Given: $\text{A}=\begin{bmatrix}1&-1&0\\2&1&3\\1&2&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&2&3\\2&1&3\\0&1&1\end{bmatrix}$ $\text{A}^\text{T}=\begin{bmatrix}1&2&1\\-1&1&2\\0&3&1\end{bmatrix}$ and $\text{B}^\text{T}=\begin{bmatrix}1&2&0\\2&1&1\\3&3&1\end{bmatrix}$ $\text{AB}=\begin{bmatrix}1&-1&0\\2&1&3\\1&2&1\end{bmatrix}\begin{bmatrix}1&2&3\\2&1&3\\0&1&1\end{bmatrix}$ $\Rightarrow\ \text{AB}=\begin{bmatrix}1-2+0&2-1+0&3-3+0\\2+2+0&4+1+3&6+3+3\\1+4+0&2+2+1&3+6+1\end{bmatrix}$$\Rightarrow\text{AB}=\begin{bmatrix}-1&1&0\\4&8&12\\5&5&10\end{bmatrix}$
$\Rightarrow(\text{AB})^\text{T}=\begin{bmatrix}-1&4&5\\1&8&5\\0&12&10\end{bmatrix}\ \dots(1)$
Now, $\text{B}^\text{T}\text{A}^\text{T}=\begin{bmatrix}1&2&0\\2&1&1\\3&3&1\end{bmatrix}\begin{bmatrix}1&2&1\\-1&1&2\\0&3&1\end{bmatrix}$ $\Rightarrow\text{B}^\text{A}\text{A}^\text{T}=\begin{bmatrix}1-2+0&2+2+0&1+4+0\\2-1+0&4+1+3&2+2+1\\3-3+0&6+3+3&3+6+1\end{bmatrix}$ $\Rightarrow\text{B}^\text{T}\text{A}^\text{T}=\begin{bmatrix}-1&4&5\\1&8&5\\0&12&10\end{bmatrix}\ \dots(2)$ $\Rightarrow(\text{AB})^\text{T}=\text{B}^\text{T}\text{A}^\text{T}$ [from eqs. (1) and (2)]

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 "5 Marks Questions" from MATRICES→MATRICES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting,
  1. 2 red balls,
  2. 2 blue balls,
  3. One red and one blue ball.
→
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$(x + y)^2 = 2axy$
→
Show that the function $f : R _0 \rightarrow R _0$, defined as $f ( x )=\frac{1}{x}$, is one-one onto, where $R _0$ is the set non-zero real numbers.
Is the result true, if the domain $R _0$ is replaced by N with co-domain being same as $R _0$ ?
→
The equation of tangent at (2, 3) on the curve $y^{2} = \text{ax}^{3} + \text{b is y = 4x - 5}. $ Find the value of a and b.
→
Let S be the set of all real numbers except -1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.
→
Show that the points whose position vectors are$\vec{\text{a}}=4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}, \vec{\text{b}}=2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-\hat{\text{j}}$ from a right triangle.
→
Verify Rolle's theorem for the following function on the indicated intervals $f(x) = x^2+ 5 x + 6 $on the interval $[-3, -2]$
→
Evalute the following integrals:
$\int\frac{\sin2\text{x}}{\text{a}\cos^2\text{x}+\text{b}\sin^2\text{x}}\text{dx}$
→
Using differentials, find the approximate values of the following:
$(33)^{\frac{1}{5}}$
→
Find the equations of the tangent and the normal to the curve $\text{y} = \frac{\text{x} - 7}{(\text{x} -2) (\text{x} -3)}$ at the point where it cuts the x-axis.
→