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Singular and Non Singualar Matrices

If A = a 8 x ...

Question

If A = $(\begin{matrix} a & b c & d \end{matrix})$ is such that A is singular matrix and $A^{2} - (a + d) A + K I = 0$ , 0 is null matrix, then K is equal to

b + c

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a + d

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ab + cd

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Solution

$G i v e n | A | = 0 \Rightarrow a d - b c = 0 a l s o A^{2} - (a + d) A = A (A - (a + d) I) = (\begin{matrix} a & b c & d \end{matrix}) (\begin{matrix} - d & b c & - a \end{matrix}) = (\begin{matrix} - a d + b c & a b - b a - c d + c d & b c - a d \end{matrix}) = (\begin{matrix} 0 & 0 0 & 0 \end{matrix}) \Rightarrow K I = 0 \Rightarrow K = 0$

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If A = (abcd) is such that A is singular matrix and A2−(a+d)A+KI=0, 0 is null matrix, then K is equal to

Given |A|=0⇒ad−bc=0also A2−(a+d)A=A(A−(a+d)I)=(abcd)(−dbc−a)=(−ad+bcab−ba−cd+cdbc−ad)=(0000)⇒KI=0⇒K=0

If A = $(\begin{matrix} a & b c & d \end{matrix})$ is such that A is singular matrix and $A^{2} - (a + d) A + K I = 0$ , 0 is null matrix, then K is equal to