Question

Need Help ASAP!!!!

Subject: Linear Algebra

(a) Let A= r 7 T 5 2 4 0 1 i. Compute det A in terms of r. ii. Find all value(s) of z such that A is NOT invertible. (b) Let the characteristic polynomial of a matrix B be – 23 +22 +6%. i. Find the size of B. ii. Find all the eigenvalues of B including multiplicity. iii. Find the determinant of B.

Answer 1

Question(a)

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$A=\begin{pmatrix}x&7&x\\ 5&2&4\\ 0&1&1\end{pmatrix}$

find determinant using first row

$det(A)= x\det \begin{pmatrix}2&4\\ 1&1\end{pmatrix}-7\cdot \det \begin{pmatrix}5&4\\ 0&1\end{pmatrix}+x\det \begin{pmatrix}5&2\\ 0&1\end{pmatrix}$

det(A) = (2 - 4) – 7(5-0) + (5-0)

$det(A)= x\left(-2\right)-7\cdot \:5+x\cdot \:5$

${\color{Red} det(A)= 3x-35}$

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when determinant of the matrix is zero then the matrix is not invertible

$3x-35=0$

$3x=35$

${\color{Red} x=\frac{35}{3}}$

so when ${\color{Red} x=\frac{35}{3}}$ then the matrix is not invertible

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Question(b)

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characteristic polynomial is

$y=-x^3+x^2+6x$

here degree of the polynomial is 3 so size of the matrix is ${\color{Red} 3 \times 3}$

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to find the eigenvalue take y=0

$-x^3+x^2+6x=0$

$-x\left(x^2-x-6\right)=0$

$-x(x^2-3x+2x-6)=0$

$-x(x(x-3)+2(x-3))=0$

$-x\left(x+2\right)\left(x-3\right)=0$

${\color{Red} x=0,\:x=-2,\:x=3}$..................eigenvalues

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As we know the determinant of a matrix is equal to the products of all eigenvalues.

$det(B)=0\cdot 2\cdot 3$

${\color{Red} det(B)=0 }$..................determinant