The determinant of a 3×3 matrix is the sum of the products of elements of any row (or column) with their corresponding cofactors. The cofactor of an element is the product of its minor with (-1) i+j where i is the row number and j is the column number containing it. Minor of an element of a 3×3 matrix is the determinant of 2×2 matrix obtained after leaving the row and column containing it. It is nothing but the difference of cross products of the elements in order. The determinant of a 2×2 matrix is defined as ad-bc. Also, the determinant of a matrix gives an important information when the elements of a matrix are associated with a linear transformation. The determinant of a matrix is used to solve the system of equations. It can be computed using a specific mathematical operation on the elements of the matrix. The determinant of a matrix M is denoted by |M| or detM.

$determinant of a matrix$

Then the function f:S->R is called the determinant. Let S be the set of all square matrices and R be the set of real numbers. Any opinions expressed on this website are entirely mine, and do not necessarily reflect the views of any of my employers.Every square matrix is associated with a real number which is called the determinant of the square matrix. All text and images on this website not specifically attributed to another source were created by me and I reserve all rights as to their use. Jeff Cruzan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. We often want to swap columns or rows in order to make a matrix triangular or block diagonal, to make the determinant easier to calculate. Note that if, in the first instance, we swapped any other two rows, the sign of the determinant would change again. Now after swapping the top two rows, notice that the determinant has only changed sign: First we'll write out a generic (no numbers) determinant: Here are examples of the effect row and column swapping on the determinant of a 3 × 3 matrix. Row & column swapping and the determinant The determinant has the same absolute value, but switches sign when either two rows or two columns are swapped. In order from top to bottom they are 3×3, 2×2, 1×1, 1×1 and 3×3 blocks.

The determinant of such a matrix is just the product of the determinants of the five blocks. Recall that a block diagonal matrix is one that has elements only symmetrically distributed in square blocks along the diagonal. The determinant of a block-diagonal matrix is the product of the determinants of the blocks.

This property means that if we can manipulate a matrix into upper- or lower-triangular form, we can easily find its determinant, even for a large matrix. Likewise, the determinant of this lower-triangular matrix is acf = adf, the product of the elements along the main diagonal. I won't try to prove this for all matrices, but it's easy to see for a 3×3 matrix:Īdf + be(0) + c(0)(0) - (0)dc - (0)ea - f(0)b The determinant of an upper-triangular or lower-triangular matrix is the product of the elements on the diagonal. For the 3×3, all of the other elements of the determinant expression except the first ( abc in this case) are zero. This is pretty easy to see using a 3×3 or 2×2 matrix. The determinant of a diagonal matrix is the product of the elements along the diagonal. We want to get to the point where we can solve for determinants of larger matrices, but first it's good to describe a few properties of determinants. Another way of saying that is: The equations are not linearly independent. And it turns out that this feature is true of matrices of any size: If the determinant of a coefficient matrix is zero, the system has no solution. So it turns out that this number a 1b 2 - a 2b 1 is quite important, important enough to get a name, the determinant. Furthermore, if a 1b 2 - a 2b 1 is zero, the system can't have a solution because we can't divide by zero. Now the thing to notice about these two expressions is that the denominators ( a 1b 2 - a 2b 1) are the same and that the terms of the denominator all come from the coefficient matrix. Now if we solve each equation for y, we get these two equations, each from one of the original linear equations: Where a 1, a 2, b 1, and b 2 are the numerical coefficients of the variables x and y. We'll begin with a 2-equations, 2-unknown (2-D) problem and look for some general (not dependent on the actual numbers) solutions.

Determinant of a matrix