Question

select the correct answer.
which matrix is invertible?
a. \\(\

$$\begin{bmatrix}-4 & 2\\\\10 & 5\\end{bmatrix}$$

\\)
b. \\(\

$$\begin{bmatrix}9 & -3\\\\-3 & 1\\end{bmatrix}$$

\\)
c. \\(\

$$\begin{bmatrix}14 & 7\\\\-4 & -2\\end{bmatrix}$$

\\)
d. \\(\

$$\begin{bmatrix}0 & 4\\\\0 & -4\\end{bmatrix}$$

\\)

Explanation:

Step1: Recall invertibility rule for 2x2 matrices

A 2x2 matrix

$$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$

is invertible if its determinant $\det = ad - bc
eq 0$.

Step2: Calculate det for Matrix A

$\det(A) = (-4)(5) - (2)(10) = -20 - 20 = -40$

Step3: Calculate det for Matrix B

$\det(B) = (9)(1) - (-3)(-3) = 9 - 9 = 0$

Step4: Calculate det for Matrix C

$\det(C) = (14)(-2) - (7)(-4) = -28 + 28 = 0$

Step5: Calculate det for Matrix D

$\det(D) = (0)(-4) - (4)(0) = 0 - 0 = 0$

Step6: Identify invertible matrix

Only Matrix A has a non-zero determinant, so it is invertible.

Answer:

A.

$$\begin{bmatrix}-4&2\\10&5\end{bmatrix}$$