Math

QuestionCari nilai x+yx + y jika matriks songsang bagi (1437)\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right) ialah 1x(743y)\frac{1}{x}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right).

Studdy Solution

STEP 1

Assumptions1. The given matrix is (1437)\left(\begin{array}{cc}1 &4 \\ -3 & -7\end{array}\right). The inverse matrix is given by 1x(743y)\frac{1}{x}\left(\begin{array}{rr}-7 & -4 \\3 & y\end{array}\right)3. We need to find the value of x+yx+y
4. The inverse of ax matrix (abcd)\left(\begin{array}{cc}a & b \\ c & d\end{array}\right) is 1adbc(dbca)\frac{1}{ad-bc}\left(\begin{array}{rr}d & -b \\ -c & a\end{array}\right)

STEP 2

First, we need to find the determinant of the given matrix. The determinant of a2x2 matrix (abcd)\left(\begin{array}{cc}a & b \\ c & d\end{array}\right) is given by adbcad-bc.
Determinant=adbcDeterminant = ad - bc

STEP 3

Now, plug in the given values for a, b, c, and d to calculate the determinant.
Determinant=(1)(7)()(3)Determinant = (1)(-7) - ()(-3)

STEP 4

Calculate the determinant.
Determinant=7(12)=Determinant = -7 - (-12) =

STEP 5

From the formula for the inverse of a2x2 matrix, we know that the determinant is the reciprocal of the factor multiplying the matrix. Therefore, xx is equal to the determinant.
x=5x =5

STEP 6

From the formula for the inverse of a2x2 matrix, we know that the element in the second row and second column of the inverse matrix is the same as the element in the first row and first column of the original matrix. Therefore, yy is equal to1.
y=1y =1

STEP 7

Now that we have the values for xx and yy, we can find the value of x+yx+y.
x+y=5+1x + y =5 +1

STEP 8

Calculate the value of x+yx+y.
x+y=5+1=6x + y =5 +1 =6The value of x+yx+y is6.

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