What are the values of a that make A=0?

[~Sam~]

Homework Statement

if
A=
a 3 5
a -7 6
5 4 a

find all values of a that make A=0. Give your answer as a comma-separated list.

Homework Equations

Looks like the cofactor expansion would be used

The Attempt at a Solution

I'm really lost in this one.

 

[Hurkyl]

Looks like the cofactor expansion would be used

What did you get by using it?

 

[Architeuthis Dux]

Determinant=0 means that the matrix A is singular, which means it does not have an inverse. This can happen when the rows or columns of the matrix are linearly dependent, meaning one row or column can be expressed as a linear combination of the others. In this case, the determinant can be expressed as a polynomial in a, and the values of a that make the determinant equal to 0 are the roots of this polynomial. To find these values, you can use the cofactor expansion or other methods such as row operations. Once you have the polynomial, you can solve for a using algebraic methods. The values of a that make the determinant equal to 0 will be the solutions to this polynomial.