Question

Find the inverse of A=
Find the Matrix equation of AX=b for X

Answer 1

`\textbf{A}^(-1) =\left[ \begin{array}{cc}-30&71\\-11&26\end{array}\right]`

`\textbf{X}=\left[\begin{array}{c}-4\\7\end{array}\right]`

Explanation:

The inverse of a matrix M is the matrix M⁻¹.

`\textsf{If}\; \textbf{M} = \left[ \begin{array}{cc}a&b\\c&d\end{array}\right] \; \textsf{then} \; \textbf{M}^(-1)=\frac{1}{\text{det} \;\textbf{M}}\left[ \begin{array}{cc}d&-b\\-c&a\end{array}\right]`

`\textsf{For a $2 * 2$ matrix} \; \textbf{M}=\left[\begin{array}{cc}a&b\\c&d\end{array}\right], \; \textsf{the determinant of} \; \textbf{M} \; \textsf{is $ad-bc$}.`

Given matrix:

`\textbf{A}=\left[\begin{array}{cc}26&-71\\11&-30\end{array}\right]`

Therefore:

`a=26, \quad b=-71, \quad c=11, \quad d=-30`

Calculate the determinant of matrix A:

`\begin{aligned}\implies \text{det}\; \textbf{A}&=26 \cdot (-30) - (-71) \cdot 11\\&=-780+781\\&=1\end{aligned}`

Therefore, the inverse of matrix A is:

`\begin{aligned}\implies \textbf{A}^(-1) & =(1)/(1)\left[ \begin{array}{cc}-30&71\\-11&26\end{array}\right]\\\\&=\left[ \begin{array}{cc}-30&71\\-11&26\end{array}\right]\end{aligned}`

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Given matrices:

`\textbf{A}=\left[\begin{array}{cc}26&-71\\11&-30\end{array}\right], \quad \textbf{B}=\left[\begin{array}{c}-49\\-132\end{array}\right]`

`\textsf{Let}\;\textbf{X}=\left[\begin{array}{c}p\\q\end{array}\right]`

If AX = B then:

`\implies\left[\begin{array}{cc}26&-71\\11&-30\end{array}\right]\left[\begin{array}{c}p\\q\end{array}\right]=\left[\begin{array}{c}-49\\-132\end{array}\right]`

`\implies\left[\begin{array}{c}7p+(-3)q\\19p+(-8)q\end{array}\right]=\left[\begin{array}{c}-49\\-132\end{array}\right]`

Therefore:

• 
  `7p-3q=-49`

• 
  `19p-8q=-132`

Solving simultaneously:

• 
  `p=-4`

• 
  `q=7`

Therefore:

`\implies \textbf{X}=\left[\begin{array}{c}-4\\7\end{array}\right]`