The transpose of a matrix is simply the matrix you get when you swap all the rows and columns. In other words, the first row becomes the first column, the second row becomes the second column, and the nth row becomes the nth column. The determinant of a transpose of a square matrix will always be equal to the determinant of the original matrix.
Read MoreCramer’s Rule is a simple rule that lets us use determinants to solve a system of equations. It tells us that we can solve for any variable in the system by calculating D_v/D, where D_v is the determinant of the coefficient matrix, with the answer column values substituted into the column representing the variable for which we’re trying to solve, and where D is the determinant of the coefficient matrix.
Read MoreSo we can simply calculate the determinant, and then, if the determinant is 0, the matrix is not invertible, so you can’t find its inverse, but if the determinant is nonzero, the matrix is invertible, so you can find its inverse.
Read MoreTo take the cross product of two vectors (a1,a2,a3) and (b1,b2,b3), we’ll set up a 3x3 matrix with i, j, and k across the first row, the components from vector a across the second row, and the components from vector b across the third row. Then we’ll evaluate the 3x3 matrix by breaking it down into determinants.
Read MoreNow that we understand what the determinant is and how to calculate it, we want to look at other properties of determinants so that we can do more with them.
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