If (where b ≠ c) and satisfies the equation A2 + kI = 0, then

Engineering

Mathematics

Introduction to Determinants

Question

If $A = [\begin{array}{cc} a & b \\ c & d \end{array}]$ (where b ≠ c) and satisfies the equation A² + kI = 0, then

a + d = 0

k = – |A|

k = |A|

None of the above

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Given, A² + kI = 0
$A^{2} = [\begin{array}{cc} a & b \\ c & d \end{array}] [\begin{array}{cc} a & b \\ c & d \end{array}] = [\begin{array}{cc} a^{2} + bc & b (a + d) \\ c (a + d) & d^{2} + bc \end{array}]$
If we take a 2 × 2 identical matrix I, then $kI = [\begin{array}{cc} k & 0 \\ 0 & k \end{array}]$
From equation A² + kI = 0, we get $A^{2} = - kI \Rightarrow A^{2} = [\begin{array}{cc} - k & 0 \\ 0 & - k \end{array}]$
a² + bc = – k ....(1)
b(a + d) = 0 ....(2)
c(a + d) = 0 ...(3)
d² + bc = – k ....(4)

From (2) and (3) we get a + d = 0 (∵ b ≠ c)
and det(A²) = k²
det(A) = ad – bc = a – a² – bc = k (from (1))

Options A and C are correct.

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