Why check if the determinant is zero before finding the inverse matrix?

Checking if the determinant is zero is necessary because the inverse matrix formula

A^{-1}=\frac{1}{\det(A)}adj A

involves dividing by the determinant. Division by zero is undefined in mathematics. Therefore, if

\det(A)=0

, the matrix is singular and does not have an inverse. Verifying

\det(A)\neq 0

ensures that the inverse exists before performing the more complex calculations required to find the adjugate matrix.

Conditions

Recall the formula for the inverse matrix using the adjugate: $A^{-1}=\frac{1}{\det(A)}adj A$ .
Identify that $\det(A)$ is in the denominator.
Conclude that if $\det(A)=0$ , the expression is undefined.
Verify $\det(A)\neq 0$ to confirm invertibility.
Proceed with calculating the adjugate only if the determinant is non-zero.

The video explicitly calculates $\det(A)=-1$ and writes $\neq 0$ . This step provides the prerequisite for using the inverse formula later.

Believing that the adjugate matrix can be divided by zero.
Thinking that checking the determinant is optional if you are confident the matrix is invertible.
Confusing the condition for invertibility with the condition for symmetry.