Question

Use determinants to find out if the matrix is invertible.[Start 3 By 3 Matrix 1st Row 1st Column 20 2nd Column 5 3rd Column 3 2nd Row 1st Column 4 2nd Column negative 15 3rd Column 6 3rd Row 1st Column 0 2nd Column 25 3rd Column negative 9 EndMatrix ]The determinant of the matrix is180180. ​(Simplify your​ answer.)Is the matrix​ invertible? Choose the correct answer below.A.The matrix is not invertible because the determinant is defined.B.The matrix is invertible because the determinant is defined.C.The matrix is not invertible because the determinant is not zero.

Answer 1

Answer:

B.The matrix is invertible because the determinant is defined.

Step-by-step explanation:

Definition: A matrix is invertible if the determinant is not equal to zero.

Given the matrix:

$A=\left(\begin{array}{ccc}20&5&3\\4&-15&6\\0&25&-9\end{array}\right)$

The determinant of the matrix:

$|A|=20\left|\begin{array}{ccc}-15&6\\25&-9\end{array}\right|-5\left|\begin{array}{ccc}4&6\\0&-9\end{array}\right|+3\left|\begin{array}{ccc}4&-15\\0&25\end{array}\right|$

$|A|=20(-15*-9-25*6)-5(-9*4-6*0)+3(4*25-0*-15)\\=20(135-150)-5(-36-0)+3(100+0)\\=20(-15)-5(-36)+3(100)\\=-300+180+300\\|A|=180$

Since the determinant, $|A|\neq 0$, the matrix is invertible because the determinant is defined.