Answer to Question #265798 in Linear Algebra for Good heart

Question #265798

Solve the following pair of linear equation by matrix method 2p +q =5


5p +3=11

1
Expert's answer
2021-11-15T16:47:21-0500

The coefficient matrix is:


A=(2150)A=\begin{pmatrix} 2 & 1 \\ 5 & 0 \end{pmatrix}

The variable matrix is:


X=(pq)X=\begin{pmatrix} p \\ q \end{pmatrix}

The constant matrix is:


B=(58)B=\begin{pmatrix} 5 \\ 8 \end{pmatrix}

Thus, to solve a system AX=B,AX=B, for X,X, multiply both sides by the inverse of AA


A1AX=A1BA^{-1}AX=A^{-1}B

and we shall obtain the solution:


X=A1BX=A^{-1}B

Provided the inverse A1A^{-1} exists, this formula will solve the system.


detA=A=2150=2(0)1(5)=50\det A=|A|=\begin{vmatrix} 2 & 1 \\ 5 & 0 \end{vmatrix}=2(0)-1(5)=-5\not=0

The inverse A1A^{-1} exists.


A1=15(0152)=(01/512/5)A^{-1}=\dfrac{1}{-5}\begin{pmatrix} 0 & -1 \\ -5 & 2 \end{pmatrix}=\begin{pmatrix} 0 & 1/5 \\ 1 & -2/5 \end{pmatrix}

X=A1B=(01/512/5)(58)X=A^{-1}B=\begin{pmatrix} 0 & 1/5 \\ 1 & -2/5 \end{pmatrix}\begin{pmatrix} 5 \\ 8 \end{pmatrix}

=(0(5)+(1/5)(8)1(5)+(2/5)(8))=(8/59/5)=\begin{pmatrix} 0(5)+(1/5)(8) \\ 1(5)+(-2/5)(8) \end{pmatrix}=\begin{pmatrix} 8/5 \\ 9/5 \end{pmatrix}

(pq)=(8/59/5)\begin{pmatrix} p \\ q \end{pmatrix}=\begin{pmatrix} 8/5 \\ 9/5 \end{pmatrix}

p=8/5,q=9/5p=8/5, q=9/5


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