# Given the two matrices, A=begin{bmatrix}1 & 2&3 1 & 1&20&1&2 end{bmatrix} text{ and } B=begin{bmatrix}1 & 1&1 2 & 1&23&1&2 end{bmatrix} (a) Find det A

Given the two matrices,
$$A=\begin{bmatrix}1 & 2&3 \\1 & 1&2\\0&1&2 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 1&1 \\2 & 1&2\\3&1&2 \end{bmatrix}$$
(a) Find det A, det B , det(AB) , det(BA) , det(5A) , $$det A^T$$ and $$det(B^6)$$
(c) Find $$A^{-1}$$ and $$B^{-1}$$ using the adjoint matrices you found in part (b)

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odgovoreh

Step 1
We have given the matrices
$$A=\begin{bmatrix}1 & 2&3 \\1 & 1&2\\0&1&2 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 1&1 \\2 & 1&2\\3&1&2 \end{bmatrix}$$
Step 2
Part(a)
Find det A:
$$det A=det\begin{bmatrix}1 & 2&3 \\1 & 1&2\\0&1&2 \end{bmatrix} =1 \cdot det\begin{bmatrix}1 & 2 \\1 & 2 \end{bmatrix}-2 \cdot det \begin{bmatrix}1 & 2 \\0 & 2 \end{bmatrix}+3 \cdot det \begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}$$
$$=1 \cdot 0 - 2 \cdot 2 +3 \cdot 1$$
=-1 Find det B:
$$det B=det\begin{bmatrix}1 & 1&1 \\2 & 1&2\\3&1&2 \end{bmatrix}$$ $$=1 \cdot det \begin{bmatrix}1 & 2 \\1 & 2 \end{bmatrix}- 1 \cdot det \begin{bmatrix}2 & 2 \\3 & 2 \end{bmatrix}+1 \cdot det \begin{bmatrix}2 & 1 \\3 & 1 \end{bmatrix}$$
$$=1 \cdot 0 - 1 \cdot (-2) +1 \cdot (-1)$$
=1
Step 3
Find det(AB) and det(BA)
According to determinant properties,
$$det(AB)=det A \times det B$$
$$=-1 \times 1$$
=-1
$$det(BA)=det B \times det A$$
$$=1 \times -1$$
=-1
Step 4
Find det(5A)
$$det(5A)=5^3 \times det A$$
$$=125 \times -1$$
=-125
Find $$det A^T:$$
$$det A^T = det A$$
=-1
Find $$det(B^6):$$
$$det(B^6)=(det B)^6$$
$$=1^6$$
=1
Step 5
Part (b)
$$A=\begin{bmatrix}1 & 2&3 \\1 & 1&2\\0&1&2 \end{bmatrix}$$
The cofactors matrix is
$$C=\begin{bmatrix}+det\begin{bmatrix}1 & 2 \\1 & 2 \end{bmatrix} & -det\begin{bmatrix}1 & 2 \\0 & 2 \end{bmatrix}& +det\begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}\\-det\begin{bmatrix}2 & 3 \\1 & 2 \end{bmatrix} & +det\begin{bmatrix}1 & 3 \\0 & 2 \end{bmatrix}&-det\begin{bmatrix}1 & 2 \\0 & 1 \end{bmatrix}\\+det\begin{bmatrix}2 & 3 \\1 & 2 \end{bmatrix}&-det\begin{bmatrix}1 & 3 \\1 & 2 \end{bmatrix}&+det\begin{bmatrix}1 & 2 \\1 & 1 \end{bmatrix} \end{bmatrix}$$
$$C=\begin{bmatrix}+(2-2) & -(2-0)&+(1-0) \\-(4-3) & +(2-0)&-(1-0)\\+(4-3)&-(2-3)&+(1-2) \end{bmatrix}$$
$$C=\begin{bmatrix}0 & -2&1 \\-1 & 2&-1\\1&1&-1 \end{bmatrix}$$
$$adj A=C^T=\begin{bmatrix}0 & -1&1 \\-2 & 2&1\\1&-1&-1 \end{bmatrix}$$
Step 6
$$B=\begin{bmatrix}1 & 1&1 \\2 & 1&2\\3&1&2 \end{bmatrix}$$
The cofactors matrix is
$$C=\begin{bmatrix}+det\begin{bmatrix}1 & 2 \\1 & 2 \end{bmatrix} & -det\begin{bmatrix}2 & 2 \\3 & 2 \end{bmatrix}& +det\begin{bmatrix}2 & 1 \\3 & 1 \end{bmatrix}\\-det\begin{bmatrix}1 & 1 \\1 & 2 \end{bmatrix} & +det\begin{bmatrix}1 & 1 \\3 & 2 \end{bmatrix}&-det\begin{bmatrix}1 & 1 \\3 & 1 \end{bmatrix}\\+det\begin{bmatrix}1 & 1 \\1 & 2 \end{bmatrix}&-det\begin{bmatrix}1 & 1 \\2 & 2 \end{bmatrix}&+det\begin{bmatrix}1 & 1 \\2 & 1 \end{bmatrix} \end{bmatrix}$$
$$C=\begin{bmatrix}+(2-2) & -(4-6)&+(2-3) \\-(2-1) & +(2-3)&-(1-3)\\+(2-1)&-(2-2)&+(1-2) \end{bmatrix}$$
$$C=\begin{bmatrix}0 & 2&-1 \\-1 & -1&2\\1&0&-1 \end{bmatrix}$$
$$adj B=C^T=\begin{bmatrix}0 & -1&1 \\2 & -1&0\\-1&2&-1 \end{bmatrix}$$
Step 7
Part (c)
Find $$A^{-1} and B^{-1}$$
$$A^{-1}=\frac{1}{det A}adj A$$
$$=\frac{1}{-1}\begin{bmatrix}0 & -1&1 \\-2 & 2&1\\1&-1&-1 \end{bmatrix}$$
$$=\begin{bmatrix}0 & 1&-1 \\2 & -2&-1\\-1&1&1 \end{bmatrix}$$
$$B^{-1}=\frac{1}{det B}adj B$$
$$=\frac{1}{1}\begin{bmatrix}0 & -1&1 \\2 & -1&0\\-1&2&-1 \end{bmatrix}$$
$$=\begin{bmatrix}0 & -1&1 \\2 & -1&0\\-1&2&-1 \end{bmatrix}$$