For the matrix . Find the values of a and b such that A2 + aA + bI = 0.

Engineering

Mathematics

Algebra of Matrices

Multiplication of Matrices

Inverse of a Matrix

Question

For the matrix $A = [\begin{array}{cc} 3 & 2 \\ 1 & 1 \end{array}]$ . Find the values of a and b such that A² + aA + bI = 0.

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$A^{2} = [\begin{array}{ll} 3 2 \\ 1 1 \end{array}] [\begin{array}{ll} 3 2 \\ 1 1 \end{array}]$

$A^{2} = [\begin{array}{ll} 11 8 \\ 4 3 \end{array}]$

A² = aA + bI

$= [\begin{array}{ll} 11 8 \\ 4 3 \end{array}] + [\begin{array}{cc} 3 a & 2 a \\ a & a \end{array}] + [\begin{array}{ll} b 0 \\ 0 b \end{array}]$

$= [\begin{array}{ll} 11 + 3 a + b 8 + 2 a \\ 4 + a 3 + a + b \end{array}] \neq [\begin{array}{ll} 0 0 \\ 0 0 \end{array}]$

4 + a = 0 ⇒ a = – 4

8 + 2a = 0 ⇒ a = – 4

if a = – 4 ⇒ 11 – 12 + b = 0

⇒ b = 1

3 – 4 + b = 0 ⇒ b = 1

∴ (0, b) ≠ (–4, 1)