If the matrix $$A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right]$$ satisfies the equation
$${A^{20}} + \alpha {A^{19}} + \beta A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 & 0 & 1 \cr
} } \right]$$ for some real numbers $\alpha$ and $\beta$, then $\beta$ $-$ $\alpha$ is equal to ___________.
Answer (integer)
4
Solution
$${A^2} = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right]\left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right] = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 & 0 & 1 \cr
} } \right]$$<br><br>$${A^3} = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 & 0 & 1 \cr
} } \right]\left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right] = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 8 & 0 \cr
3 & 0 & { - 1} \cr
} } \right]$$<br><br>$${A^4} = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 & 0 & 1 \cr
} } \right]\left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 & 0 & 1 \cr
} } \right] = \left[ {\matrix{
1 & 0 & 0 \cr
0 & {16} & 0 \cr
0 & 0 & 1 \cr
} } \right]$$<br><br>$$\eqalign{
& . \cr
& . \cr
& . \cr
& . \cr
& . \cr
& . \cr} $$<br><br>$${A^{19}} = \left[ {\matrix{
1 & 0 & 0 \cr
0 & {{2^{19}}} & 0 \cr
3 & 0 & { - 1} \cr
} } \right],{A^{20}} = \left[ {\matrix{
1 & 0 & 0 \cr
0 & {{2^{20}}} & 0 \cr
0 & 0 & 1 \cr
} } \right]$$<br><br>$L.H.S. = {A^{20}} + \alpha {A^{19}} + \beta A =$
<br><br>$$\left[ {\matrix{
{1 + \alpha + \beta } & 0 & 0 \cr
0 & {{2^{20}} + \alpha {2^{19}} + 2\beta } & 0 \cr
{3\alpha + 3\beta } & 0 & {1 - \alpha - \beta } \cr
} } \right]$$<br><br>$$R.H.S. = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 & 0 & 1 \cr
} } \right] $$
<br><br>$\Rightarrow \alpha + \beta = 0$ and ${2^{20}} + \alpha {2^{19}} + 2\beta = 4$<br><br>$\Rightarrow {2^{20}} + \alpha ({2^{19}} - 2) = 4$<br><br>$\Rightarrow \alpha = {{4 - {2^{20}}} \over {{2^{19}} - 2}} = - 2$<br><br>$\Rightarrow \beta = 2$<br><br>$\therefore$ $\beta - \alpha = 4$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
This question is part of PrepWiser's free JEE Main question bank. 274 more solved questions on Matrices and Determinants are available — start with the harder ones if your accuracy is >70%.