Consider the system of equations x1 + 2x2 = −1 2x1 + 5x2 = −10. (a) Use the algorithm to find the inverse of the coefficient matrix A. (b) Use the inverse to calculate the solution of the system.

Question

Consider the system of equations

[latex]x_{1}+2 x_{2}=-1[/latex]
[latex]2 x_{1}+5 x_{2}=-10[/latex].

(a) Use the algorithm to find the inverse of the coefficient matrix A.

(b) Use the inverse to calculate the solution of the system.

Accepted Answer

(a) We begin by forming the (2 × 4) matrix [A|I],
[latex][A \mid I]=\left[\begin{array}{llll}1 & 2 & 1 & 0 \2 & 5 & 0 & 1\end{array} ight][/latex].
We next row reduce [A|I] to [I |B] as follows:
[latex]R_{2}-2 R_{1}:[/latex]
[latex]\left[\begin{array}{rrrr}1 & 2 & 1 & 0 \0 & 1 & -2 & 1\end{array} ight][/latex]
[latex]R_{1}-2 R_{2}:[/latex]
[latex]\left[\begin{array}{rrrr}1 & 0 & 5 & -2 \0 & 1 & -2 & 1\end{array} ight][/latex].
Thus, [latex]A^{-1}[/latex] is the matrix
[latex]\left[\begin{array}{rr}5 & -2 \-2 & 1\end{array} ight][/latex].
(b) The solution to the system is [latex]x =A^{-1} b[/latex] where
[latex]b =\left[\begin{array}{r}-1 \-10\end{array} ight][/latex].
Now, [latex] A^{-1} b =[15,-8]^{T}[/latex], so the solution is [latex]x_{1}=15, x_{2}=-8[/latex].

Question 1.9.3: Consider the system of equations x1 + 2x2 = −1 2x1 + 5x2 = −...

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