Consider the following matrix. $$ A=\left[\begin{array}{cc}7 & 3 \ -5 & -2\end{array}\right] $$ Choose the correct description of $$ A $$. Find $$ A^{-1} $$ if it exists. $$ A \text { is nonsingular. That is, it has an inverse. } $$ $$ A^{-1}=\square $$ is singular. That is, its inverse doesn't exist.

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To determine if the matrix $$ A $$ is nonsingular and to find its inverse, we need to calculate the determinant of $$ A $$ and then use the formula for the inverse of a 2x2 matrix if the determinant is non-zero.

The determinant of a 2x2 matrix $$ A = \left[\begin{array}{cc}a & b \ c & d\end{array}\right] $$ is given by $$ ad - bc $$.

For matrix $$ A $$, we have:
$$ a = 7, \quad b = 3, \quad c = -5, \quad d = -2 $$

So, the determinant of $$ A $$ is:
$$ \det(A) = (7)(-2) - (3)(-5) = -14 + 15 = 1 $$

Since the determinant is non-zero (it is 1), the matrix $$ A $$ is nonsingular, and it has an inverse.

The inverse of a 2x2 matrix $$ A = \left[\begin{array}{cc}a & b \ c & d\end{array}\right] $$ is given by:
$$ A^{-1} = \frac{1}{\det(A)} \left[\begin{array}{cc}d & -b \ -c & a\end{array}\right] $$

Using the values from matrix $$ A $$, we get:
$$ A^{-1} = \frac{1}{1} \left[\begin{array}{cc}-2 & -3 \ 5 & 7\end{array}\right] $$
$$ A^{-1} = \left[\begin{array}{cc}-2 & -3 \ 5 & 7\end{array}\right] $$

Therefore, the correct description of $$ A $$ is that it is nonsingular, and the inverse of $$ A $$ is:
$$ A^{-1} = \left[\begin{array}{cc}-2 & -3 \ 5 & 7\end{array}\right] $$

The statement "That is, its inverse doesn't exist" is incorrect because we have just shown that $$ A^{-1} $$ does exist.
Matrix $$ A $$ is nonsingular with an inverse $$ A^{-1} = \left[\begin{array}{cc}-2 & -3 \ 5 & 7\end{array}\right] $$.

Consider the following matrix.

Choose the correct description of .
Find if it exists.

is singular. That is, its inverse doesn’t exist.

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Consider the following matrix. Choose the correct description of . Find if it exists. is singular. That is, its inverse doesn’t exist.