If $A = |\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}|, f i n d A^{- 1} .$ Use it solve the system of equations.
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y- 2z =-3

Solution

$A = [\begin{matrix} 2 & - 3 & 5 \\ 3 & 3 & - 4 \\ 1 & 1 & - 2 \end{matrix}] ∴ | A | = 2 (- 4 + 4) + 3 (- 6 + 4) + 5 (3 - 2) = 0 - 6 + 5 = - 1 \neq 0$

Now, A₁₁ 0, A₁₂ = 2., A₁₂ =1

A₂₁ = -1, A₂₂ = -9, A₂₃ = -5

A₃₁ = 2, A₃₂ = 23, A₃₃ = 13

$∴ A^{- 1} = \frac{1}{| A |} (a d j A) = - [\begin{matrix} 0 & - 1 & 2 \\ 2 & - 9 & 23 \\ 1 & - 5 & 13 \end{matrix}] = [\begin{matrix} 0 & 1 & - 2 \\ - 2 & 9 & - 23 \\ - 1 & 5 & - 13 \end{matrix}] . . . (1)$

Now, the given system of equations can be written in the form of AX = B, where

$A = [\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}], X = [\begin{matrix} x \\ y \\ z \end{matrix}] and B = [\begin{matrix} 11 \\ - 5 \\ - 3 \end{matrix}] The solution of the system of equations is given by X = A^{- 1} B X = A^{- 1} B [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 0 & 1 & - 2 \\ - 2 & 9 & - 23 \\ - 1 & 5 & - 13 \end{matrix}] [\begin{matrix} 11 \\ - 5 \\ - 3 \end{matrix}] (using 1) = [\begin{matrix} 0 - 5 + 6 \\ - 22 - 45 + 39 \\ - 11 - 25 + 39 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

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Matrices

If $A = |\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}|, f i n d A^{- 1} .$ Use it solve the system of equations.
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y- 2z =-3

Some More Questions From Matrices Chapter

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

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For Daily Free Study Material Join wiredfaculty

Matrices

If A = 2-3532-411-2, find A-1. Use it solve the system of equations.2x - 3y + 5z = 113x + 2y - 4z = -5x + y- 2z =-3

Some More Questions From Matrices Chapter

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

If $A = |\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}|, f i n d A^{- 1} .$ Use it solve the system of equations.
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y- 2z =-3