Question

Given 4 2
-1 1
M = 6I,where M is a matrix and I is a unit matrix of order 2x2.

State the order of matrix M and find the matrix M.

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Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

$\displaystyle\begin{pmatrix} 4 & 2\\ - 1 & 1 \end{pmatrix} \sf{M = 6I}$

where M is a matrix and

I is a unit matrix of order 2x2

TO DETERMINE

• Order of matrix M

• The matrix M

CALCULATION

It is given that

$\displaystyle\begin{pmatrix} 4 & 2\\ - 1 & 1 \end{pmatrix} \sf{M = 6I} \: \: \: \: ...(1)$

$\sf{Let \: \: A} = \displaystyle\begin{pmatrix} 4 & 2\\ - 1 & 1 \end{pmatrix}$

Then det A = 4 + 2 = 6

$\sf{Since \: \: \: det \: A \ne \: 0}$

$\sf{So \: \: { A}^{ - 1} \: \: exists}$

Now

$\sf{ adj \: A} = \displaystyle\begin{pmatrix} 1 & - 2\\ 1 & 4 \end{pmatrix}$

So

$\sf{{ A }^{ - 1} } = \displaystyle \sf{\frac{adj \:A }{det \:A } } = \frac{1}{6} \begin{pmatrix} 1 & - 2\\ 1 & 4 \end{pmatrix}$

From Equation (1) we get

$\sf{AM = 6I \: }$

$\implies \: \sf{M = {A}^{ - 1} \times 6I \: }$

$\implies \: \sf{ M}= \frac{1}{6} \times 6 \times \begin{pmatrix} 1 & - 2\\ 1 & 4 \end{pmatrix} \times \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$

$\implies \: \sf{ M}= \begin{pmatrix} 1 & - 2\\ 1 & 4 \end{pmatrix}$

RESULT

$\boxed{ \: \: \: \sf{ M}= \begin{pmatrix} 1 & - 2\\ 1 & 4 \end{pmatrix} \: \: \: }$

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LEARN MORE FROM BRAINLY

For a square matrix A and a non-singular matrix B of the same order, the value of det(B^-1 AB)

https://brainly.in/question/23783093

Answer 2

Answer:

Let M= [ a b , c d ]

[4 -1 , 2 1] [a b , c d] = 6 [ 1 0 , 0 1 ]

4a + 2b -a+b , 4c +2d ] =[ 6 0, 0 6]

b =a

4a + 2a =6

6a=6

a= 1

4×1+2b =6

4+2b =6

2b =6-4

b = 1

-c+d=0

d=6+c

4c+2(6+c) = 0

4c+12+2c=0

6c=-12

c=-2

4(-2) +2d =0

-8+2d=0

2d=8

d=4

M= [1 1 , -2 4]

order of M = 2×2

HOPE IT HELPED