Find k, if A= [ ll 3 & -2 4 & -2 ] and A^2=k A-2 l. - Vidyadip

Question

Find $k$, if $A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]$ and $A^2=k A-2 l$.

✓

Answer

$
\begin{aligned}
A^2= A \cdot A & =\left[\begin{array}{ll}
3 & -2 \\
4 & -2
\end{array}\right]\left[\begin{array}{ll}
3 & -2 \\
4 & -2
\end{array}\right] \\
& =\left[\begin{array}{rr}
9-8 & -6+4 \\
12-8 & -8+4
\end{array}\right]=\left[\begin{array}{ll}
1 & -2 \\
4 & -4
\end{array}\right] \\
k A -2 I & =k\left[\begin{array}{ll}
3 & -2 \\
4 & -2
\end{array}\right]-2\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
3 k & -2 k \\
4 k & -2 k
\end{array}\right]-\left[\begin{array}{ll}
2 & 0 \\
0 & 2
\end{array}\right] \\
& =\left[\begin{array}{cc}
3 k-2 & -2 k \\
4 k & -2 k-2
\end{array}\right]
\end{aligned}
$
But, $A ^2=k A -2 I$
$
\therefore\left[\begin{array}{rr}
1 & -2 \\
4 & -4
\end{array}\right]=\left[\begin{array}{rr}
3 k-2 & -2 k \\
4 k & -2 k-2
\end{array}\right]
$
By equality of matrices,
$
\begin{aligned}
& 1=3 k-2 \ldots \\
& -2=-2 k \ldots \ldots \\
& 4=4 k \ldots \ldots . \\
& -4=-2 k-2
\end{aligned}
$
From (2), $k =1$. $k =1$ also satisfies equation (1), (3) and (4). Hence, $k=1$.

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 "Solve the Following Question.(4 Marks)" from Matrices (p-1)→Matrices (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Find $\frac{ d y}{ d x}$, if $y =\sqrt[5]{\left(3 x^2+8 x+5\right)^4}$

The probability that a bulb produced by a factory will use fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.

The equation of the two lines of regression are 3x + 2y – 26 = 0 and 6x + y – 31 = 0. Find
(i) Means of X and Y
(ii) Correlation coefficient between X on Y
(iii) Estimate of Y for X = 2
(iv) var (X) if var (Y) = 36

Obtain the trend values for the data, using 3-yearly moving averages

Year	1976	1977	1978	1979	1980	1981
Production	0	4	4	2	6	8
Year	1982	1983	1984	1985	1986
Production	5	9	4	10	10

Solve the following differential equation $y x \frac{ d y}{ d x}= x ^2+2 y ^2$

Find the probability of the number of successes in two tosses of a die, where success is defined as (i) number greater than 4 (ii) six appearing in at least one toss.

If $A=\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right], B=\left[\begin{array}{cc}2 & 1 \\ 4 & -1 \\ -3 & 3\end{array}\right], C=\left[\begin{array}{cc}1 & 2 \\ -1 & 4 \\ -2 & 3\end{array}\right]$, then show that
(i) $(A+B)^{\top}=A^{\top}+B^{\top}$
(ii) $(A-C)^{\top}=A^{\top}-C^{\top}$

For certain X and Y series, which are correlated the two lines of regression are 10y = 3x + 170 and 5x + 70 = 6y. Find the correlation coefficient between them. Find the mean values of X and Y.

The total cost of manufacturing x articles is $C = 47x + 300x^2 – x^4$. Find x, for which average cost is
(i) increasing
(ii) decreasing.

After deducting commission at $7 \frac{1}{2} \%$ on first $\text{₹}$ 50,000$ and $5 \%$ on the balance of sales made by him, an agent remits $\text{₹}$ 93,750 to his principal. Find the value of goods sold by him.