### Instruction: If $\det \mathbf{A} = -1,$ then find $\det (\mathbf{7A}).$.### Response: This problem asks me to find the determinant of a matrix that is a multiple of another matrix whose determinant I know.
I recall that there is a property of determinants that relates the determinant of a multiple of a matrix to the determinant of the original matrix.
It is $\det (k\mathbf{A}) = k^n \det (\mathbf{A}),$ where $k$ is any scalar and $n$ is the size of the matrix.
In this case, $k = 7$ and $n$ is not given, but it must be the same as the size of $\mathbf{A}.$
Therefore, I can apply the property and write $\det (\mathbf{7A}) = 7^n \det (\mathbf{A}).$
Since I know $\det \mathbf{A} = -1,$ I can substitute it and get $\det (\mathbf{7A}) = 7^n (-1).$
This is the final answer, but it depends on the value of $n.$
If I had more information about the matrix $\mathbf{A},$ I could find $n$ and give a numerical value for the determinant.