Let A= [ cc 2 & -1 0 & 2 ]. If B=I- ^3 C_1(adj A +C_2(adj A)^2 - ^3 C_3(adj A)^3, then the sum of all elements of the matrix B is

Let A= [ cc 2 & -1 0 & 2 ]. If B=I- ^3 C_1(adj A +C_2(adj A)^2 - …

MCQ

Let $A=\left[\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}\right]$. If $B=I-{ }^3 C_1(\operatorname{adj} A +C_2(\operatorname{adj} A)^2$ $-{ }^3 C_3(\operatorname{adj} A)^3$, then the sum of all elements of the matrix $B$ is

A
-1
B
-3
C
-4
✓
-5

✓

Answer

Correct option: D.

-5

(d) : $A=\left[\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}\right]$, then $\operatorname{adj} A=\left[\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right]$
Given, $B=I-{ }^3 C _1(\operatorname{adj} A)+{ }^3 C _2(\operatorname{adj} A)^2-{ }^3 C _3(\operatorname{adj} A)^3$
Then, $B=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]-3\left[\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right]+3\left[\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right]\left[\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right]$
$
-\left[\begin{array}{ll}
2 & 1 \\
0 & 2
\end{array}\right]\left[\begin{array}{ll}
2 & 1 \\
0 & 2
\end{array}\right]\left[\begin{array}{ll}
2 & 1 \\
0 & 2
\end{array}\right]
$
$
\begin{aligned}
& =\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]-\left[\begin{array}{ll}
6 & 3 \\
0 & 6
\end{array}\right]+3\left[\begin{array}{ll}
4 & 4 \\
0 & 4
\end{array}\right]-\left[\begin{array}{ll}
4 & 4 \\
0 & 4
\end{array}\right]\left[\begin{array}{ll}
2 & 1 \\
0 & 2
\end{array}\right] \\
& =\left[\begin{array}{cc}
-5 & -3 \\
0 & -5
\end{array}\right]+\left[\begin{array}{cc}
12 & 12 \\
0 & 12
\end{array}\right]-\left[\begin{array}{cc}
8 & 12 \\
0 & 8
\end{array}\right]=\left[\begin{array}{cc}
-1 & -3 \\
0 & -1
\end{array}\right]
\end{aligned}
$
Sum of elements of $B=-1-3-1=-5$

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