Download App

Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants3 Marks
Question
For the matrix $A = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 1&1 \end{array}} \right]$, find the numbers a and b such that $A^2 + aA + bI = 0.$
✓

Answer

Given: $A = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 1&1 \end{array}} \right]$$\therefore {A^2} = A.A = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 3&2 \\ 1&1 \end{array}} \right]$
$= \left[ {\begin{array}{*{20}{c}} {9 + 2}&{6 + 2} \\ {3 + 1}&{2 + 1} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {11}&8 \\ 4&3 \end{array}} \right]$
$\therefore {A^2} + aA + b{I_2} = 0$
$\Rightarrow \left[ {\begin{array}{*{20}{c}} {11}&8 \\ 4&3 \end{array}} \right] + a\left[ {\begin{array}{*{20}{c}} 3&2 \\ 1&1 \end{array}} \right] + b\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right] = 0$
$\Rightarrow \left[ {\begin{array}{*{20}{c}} {11}&8 \\ 4&3 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {3a}&{2a} \\ a&a \end{array}} \right] $ $+ \left[ {\begin{array}{*{20}{c}} b&0 \\ 0&b \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&0 \\ 0&0 \end{array}} \right]$
$\Rightarrow \left[ {\begin{array}{*{20}{c}} {11 + 3a + b}&{8 + 2a + 0} \\ {4 + a + 0}&{3 + a + b} \end{array}} \right] $ $= \left[ {\begin{array}{*{20}{c}} 0&0 \\ 0&0 \end{array}} \right]$
$\therefore$ We have $11 + 3a + b = 0….(i)$
$8 + 2a + 0 = 0 ...(ii)$
$\Rightarrow 2a = - 8$
$\Rightarrow a = - 4$
Here $a = -4$ satisfies $4 + a + 0 = 0$ also, therefore $a = -4$
Putting $a = -4$ in eq. (i), $11 - 12 + b = 0$ $ \Rightarrow b - 1 = 0 \Rightarrow b = 1$
Here also $b = 1$ satisfies $3 + a + b = 0,$ therefore $b = 1$
Therefore, $a = - 4$ and $b = 1$

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 "3 Marks Question" from Determinants→Determinants — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$xy = c^2$
→
A factory has two machines $A$ and $B$. Past records show that the machine $A$ produced $60\%$ of the items of output and machine $B$ produced $40\%$ of the items. Further $2\%$ of the items produced by machine $A$ were defective and $1\%$ produced by machine $B$ were defective. If an item is drawn at random$,$ what is the probability that it is defective?
→
Check the commutativity and associativity of the following binary operations:
'*' on $N$, de fined by $a * b = a^b$ for all $a, b \in N$.
→
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\sin\text{x }\text{dx}$
→
Express $\overrightarrow{\text{AB}}$ in terms of unit vectors $\hat{\text{i}}\text{ and }\hat{\text{j}}$, when the point is:A(4, -1), B(1, 3)
Find $\Big|\overrightarrow{\text{AB}}\Big|$
→
Evaluate $\int\frac{1}{\text{x}(1+\log\text{x})}\text{ dx}$
→
Evaluate the following integrals:
$\frac{\text{x}+1}{\text{x}(1+\text{xe}^\text{x})}\ \text{dx}$
→
If $\theta$ is the angle between two unit vectors $\hat{a}$ and $\hat{b}$ then show that $\cos \frac{\theta}{2}=\frac{1}{2}|\hat{a}+\hat{b}|$.
→
Prove that $f(x) = ax + b,$ where $a, b$ are constants and $a > 0$ is an increasing function on $R.$
→
Find the mean and standard deviation of the following probability distributions:
$x_i$ 2 3 4
$p_i$ 0.2 0.5 0.3
→