If A = $[\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}]$ , verify that A² – 4A – 5I = 0.

Solution

$A^{2} = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}] = [\begin{matrix} 1 x 1 + 2 x 2 + 2 x 2 & 2 x 1 + 2 x 1 + 2 x 2 & 1 x 2 + 2 x 2 + 2 x 1 \\ 2 x 1 + 1 x 2 + 2 x 2 & 2 x 2 + 1 x 1 + 2 x 2 & 2 x 2 + 1 x 2 + 2 x 1 \\ 2 x 1 + 2 x 2 + 1 x 2 & 2 x 2 + 2 x 1 + 1 x 2 & 2 x 2 + 2 x 2 + 1 x 1 \end{matrix}] = [\begin{matrix} 1 + 4 + 4 & 2 + 2 + 4 & 2 + 4 + 2 \\ 2 + 2 + 4 & 4 + 1 + 4 & 4 + 2 + 2 \\ 2 + 4 + 2 & 4 + 2 + 2 & 4 + 4 + 1 \end{matrix}] = [\begin{matrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{matrix}]$

$4 A = 4 [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}] = [\begin{matrix} 1 X 4 & 2 X 4 & 2 X 4 \\ 2 X 4 & 1 X 4 & 2 X 4 \\ 2 X 4 & 2 X 4 & 1 X 4 \end{matrix}] = [\begin{matrix} 4 & 8 & 8 \\ 8 & 4 & 8 \\ 8 & 8 & 4 \end{matrix}]$

$5 I = 5 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{matrix}]$

$A^{2} - 4 A - 5 I = [\begin{matrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{matrix}] - 4 [\begin{matrix} 4 & 8 & 8 \\ 8 & 4 & 8 \\ 8 & 8 & 4 \end{matrix}] - [\begin{matrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{matrix}] = [\begin{matrix} 9 - 4 - 5 & 8 - 8 - 0 & 8 - 8 - 0 \\ 8 - 8 - 0 & 9 - 4 - 5 & 8 - 8 - 0 \\ 8 - 8 - 0 & 8 - 8 - 0 & 9 - 4 - 5 \end{matrix}] = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] = 0 = R . H . S .$

Some More Questions From Matrices Chapter

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Matrices

If A = $[\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}]$ , verify that A² – 4A – 5I = 0.

Some More Questions From Matrices Chapter

If a matrix has 8 elements, what arc the possible orders it can have ?

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

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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NCERT Sample Papers

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Matrices

If A = 1 2 2212221, verify that A2 – 4A – 5I = 0.

Some More Questions From Matrices Chapter

If a matrix has 8 elements, what arc the possible orders it can have ?

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

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 = $[\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}]$ , verify that A² – 4A – 5I = 0.