In order to determine if a matrix is an invertible square matrix, or a square matrix with an inverse, we can use determinants. The only matrix with a nonzero determinant is an invertible square matrix. An invertible square matrix represents a system of equations with a regular solution, and a non-invertible square matrix can represent a system of equations with no or infinite solutions.
I want to talk a little bit more about the inverse of a matrix because we can use determinants to find out if a matrix is actually invertable turns out that we have this result, a square matrix a is invertible if and only if its determinant is not 0 that the determinant of a square matrix is 0 it's not invertible so let's check this one, is matrix a equals 6,-9 10,-15 invertible. We calculate the determinant of a and that's 6 times -15 -90 minus -9 times 10 -90 so -90+90 that's 0, No a is not invertible. Now another thing another relationship between determinants and inverses is for 2 by 2 matrices I can get the inverse really easily using this formula. If our matrix has this these entries a, b, c and d then its inverse is 1 over the determinant of matrix a times this matrix and notice the difference a and d have switched and b and c have taken their opposites so along the main diagonal you just switch the the entries on the main diagonal and you take the opposites of the other entries. Don't forget to multiply by 1 over the determinant so let's let's now use that. Here's a matrix a 2,3 3,6 find a inverse, and by the way if you happen to get determinant is 0 then it doesn't have an inverse so you can also say not invertible, but first we calculate determinant of a. We get 2 times 6 12 minus 3 times 3 9 so that's 3 and so this part of the formula is going to be 1 over 3 one third so a inverse equals one third and then I take the entries, these diagonal entries I switch them 6 and 2 and these guys I take their opposites -3 -3 and you should simplify this so I'm going to multiply the one third through and I get one third times 6 2 one third times -3 -1 -1 and two thirds and there it is so really easy formula for the inverse of a 2 by 2 matrix unfortunately we don't have an easy formula like this for 3 by 3 and higher but you can use this formula from now on to find the inverse of the 2 by 2 matrix.