The matrix A²+4 A-5 I , where I is identity matrix and (A= [ array cc 1 & 2 4 & -3 array ]), equals :

Correct answer: A. (4 [ array ll 2 & 1 2 & 0 array ]) ( aligned & A^2+4 A-5 I=A × A+4 A-5 I = & [ array cc 1 & 2 4 & -3 array ] ×[ array cc 1 & 2 4 & -3 array ]+4[ array cc 1 & 2 4 & -3 array ]-5[ array ll 1 & 0 0 & 1 array ] = & [ array cc 9 & -4 -8 & 17 array ]+[ array cc 4 & 8 16 & -12 array ]-[ array cc 5 & 0 0 & 5 array ] = & [ array cc 9+4-5 & -4+8-0 -8+16-0 & 17-12-5 array ]=[ array cc 8 & 4 8 & 0 array ] = & 4[ array cc 2 & 1 2 & 0 array ] aligned )

BITSAT 2011

Exam: BITSAT
Year: 2011
Subject: Mathematics
Chapter: Matrices

BITSAT 2011 · Mathematics · Matrices

The matrix A²+4 A-5 I , where I is identity matrix and (A= [ array...

The matrix

A²+4 A-5 I

, where

I

is identity matrix and (A=

\left[

\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}

\right]), equals :

Options

A(4 $\left[$ $\begin{array}{ll}2 & 1 \\ 2 & 0\end{array}$ $\right])
B(4 $\left[$ $\begin{array}{cc}0 & -1 \\ 2 & 2\end{array}$ $\right])
C(32 $\left[$ $\begin{array}{ll}2 & 1 \\ 2 & 0\end{array}$ $\right])
D(32 $\left[$ $\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}$ $\right])

Correct answer

A. (4 $\left[$ $\begin{array}{ll}2 & 1 \\ 2 & 0\end{array}$ $\right])

Step-by-step solution

(

\begin{aligned} & A^2+4 A-5 I=A \times A+4 A-5 I \\ = & {\left[\begin{array}{cc} 1 & 2 \\ 4 & -3 \end{array}\right] \times\left[\begin{array}{cc} 1 & 2 \\ 4 & -3 \end{array}\right]+4\left[\begin{array}{cc} 1 & 2 \\ 4 & -3 \end{array}\right]-5\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] }\\ = & {\left[\begin{array}{cc} 9 & -4 \\ -8 & 17 \end{array}\right]+\left[\begin{array}{cc} 4 & 8 \\ 16 & -12 \end{array}\right]-\left[\begin{array}{cc} 5 & 0 \\ 0 & 5 \end{array}\right] }\\ = & {\left[\begin{array}{cc} 9+4-5 & -4+8-0 \\ -8+16-0 & 17-12-5 \end{array}\right]=\left[\begin{array}{cc} 8 & 4 \\ 8 & 0 \end{array}\right] }\\ = & 4\left[\begin{array}{cc}2 & 1 \\ 2 & 0 \end{array}\right] \end{aligned}

)

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