KCET · Maths · Matrices
The characteristic equation of a matrix \(\mathrm{A}\) is \(\lambda^{3}-5 \lambda^{2}-3 \lambda+2=0\), then \(\mid\) adj (A) \(\mid\) is equal to
- A 9
- B 25
- C \(\frac{1}{2}\)
- D 4
Answer & Solution
Correct Answer
(D) 4
Step-by-step Solution
Detailed explanation
We know that, \(\left[\begin{array}{ccc}0 & 0 & -\mathrm{c} \\ 1 & 0 & -\mathrm{b} \\ 0 & 1 & -\mathrm{a}\end{array}\right]\) iff
\[
\lambda^{3}+\mathrm{a} \hat{\lambda}^{2}+\mathrm{b} \lambda+\mathrm{c}=0
\]
But given that,
\(\therefore\)
\(\lambda^{3}-5 \lambda^{2}-3 \lambda+2=0\)
\(A=\left[\begin{array}{rrr}0 & 0 & -2 \\ 1 & 0 & 3 \\ 0 & 1 & 5\end{array}\right]\)
\(|A|=\left[\begin{array}{rrr}0 & 0 & -2 \\ 1 & 0 & 3 \\ 0 & 1 & 5\end{array}\right]\)
\(=0+0-2(1-0)=-2\)
\(\therefore \mid\) adj \(\left.\mathrm{A}|=| \mathrm{A}\right|^{2}=(-2)^{2}=4\)
\[
\lambda^{3}+\mathrm{a} \hat{\lambda}^{2}+\mathrm{b} \lambda+\mathrm{c}=0
\]
But given that,
\(\therefore\)
\(\lambda^{3}-5 \lambda^{2}-3 \lambda+2=0\)
\(A=\left[\begin{array}{rrr}0 & 0 & -2 \\ 1 & 0 & 3 \\ 0 & 1 & 5\end{array}\right]\)
\(|A|=\left[\begin{array}{rrr}0 & 0 & -2 \\ 1 & 0 & 3 \\ 0 & 1 & 5\end{array}\right]\)
\(=0+0-2(1-0)=-2\)
\(\therefore \mid\) adj \(\left.\mathrm{A}|=| \mathrm{A}\right|^{2}=(-2)^{2}=4\)
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