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A) Find the inverse of the matrix [5 -3; -3 4]. b) Hence solve the simultaneous equation using the matrix method. 8x+6y=12, 6x+10y=10

August 17, 2026Mathematics

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Ili kutatua milinganyo shirikishi kwa kutumia njia ya matriki, tunafuata hatua hizi:

Andika milinganyo katika mfumo wa matriki.
Milinganyo ni:
$8x+6y=12$
$6x+10y=10$

Hii inaweza kuandikwa kama $AX=B$ :
$\begin{bmatrix} 8 & 6 \\ 6 & 10 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 12 \\ 10 \end{bmatrix}$
Hapa, $A = \begin{bmatrix} 8 & 6 \\ 6 & 10 \end{bmatrix}$ , $X = \begin{bmatrix} x \\ y \end{bmatrix}$ , na $B = \begin{bmatrix} 12 \\ 10 \end{bmatrix}$ .
Tafuta kinyume cha matriki $A$ ( $A^{-1}$ ).
Kwa matriki $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , kinyume chake ni $A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ .

Kwanza, tafuta determinant ya $A$ :
$\det(A) = (8)(10) - (6)(6) = 80 - 36 = 44$
Sasa, tafuta kinyume cha $A$ :
$A^{-1} = \frac{1}{44} \begin{bmatrix} 10 & -6 \\ -6 & 8 \end{bmatrix}$
Tatua kwa $X$ kwa kutumia $X = A^{-1}B$ .
$\begin{bmatrix} x \\ y \end{bmatrix} = \frac{1}{44} \begin{bmatrix} 10 & -6 \\ -6 & 8 \end{bmatrix} \begin{bmatrix} 12 \\ 10 \end{bmatrix}$
Fanya kuzidisha kwa matriki:
$\begin{bmatrix} x \\ y \end{bmatrix} = \frac{1}{44} \begin{bmatrix} (10)(12) + (-6)(10) \\ (-6)(12) + (8)(10) \end{bmatrix}$ $\begin{bmatrix} x \\ y \end{bmatrix} = \frac{1}{44} \begin{bmatrix} 120 - 60 \\ -72 + 80 \end{bmatrix}$ $\begin{bmatrix} x \\ y \end{bmatrix} = \frac{1}{44} \begin{bmatrix} 60 \\ 8 \end{bmatrix}$
Zidisha kwa $\frac{1}{44}$ :
$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{60}{44} \\ \frac{8}{44} \end{bmatrix}$
Rahisisha majibu.
$x = \frac{60}{44} = \frac{15}{11}$ $y = \frac{8}{44} = \frac{2}{11}$
Kwa hiyo, majibu ni:
$\boxed{x = \frac{15}{11}, y = \frac{2}{11}}$