What is a determinant?

1. What is a determinant? A “useful number” associated with an nn matrix. We’ll learn how to find determinants of 2  2 and 3  3 matrices and then see one of the many uses. example: = - 8 - (-3) = - 5 The determinant or “useful” number associated with the example matrix is - 5.

2. cofactor minor What all of this says it that you can choose any row (the ith row) of a matrix or any column (the jth column) of a matrix and find the determinant with the formulas given. This is called expanding by cofactors.

3. First let’s see how to compute the minors and cofactors and then we’ll see how to use the formula. Mij = The determinant of what’s left when you cross off an element’s row and column. Mij = The determinant of what’s left when you cross off an element’s row and column. Mij = The determinant of what’s left when you cross off an element’s row and column. = 2 - 0 = 2 Find M31 = - 2 - (-6) = 4 3rd row Find M12 1st column 1st row Now we can find minors, we need to see how we use this to get cofactors. 2nd column determinant of what’s left we’ll be finding the minor for this element

4. The cofactor is the minor with a sign change considered If i + j is even then -1 to an even power will just be 1. If i + j is odd, -1 to an odd power is -1. i is the row number and j is the column number. The pattern of when you get a positive and when you get a negative is shown at the right. The cofactor is just the minor for the position but when it falls in a negative position you change the sign.

5. Okay----we are ready to use the formula: This says that we can pick any row or any column in the matrix and multiply the elements of the matrix in that row or column by their cofactors and find the determinant.

6. because this falls in a negative position Let’s choose the first column to expand by cofactors to find the determinant (I just randomly chose a column). Then: We need A11 = - 4 - 12 = -16 = - 43 A21 = - (- 10 - 21) = 31 A31 = 20 - 14 = 6 Now we multiply all elements in the first column by their cofactors and add them up to get the determinant.

7. Properties of Determinants P1. If you trade rows places you must change the sign of the determinant. P2. If you multiply a row by a number you must divide the determinant by that number. P3. If you add a multiple of one row to another row, it doesn’t affect the determinant. P5. If two rows (or columns) of A are the same, then det(A)=0 P6. If a row (or column) of A is all zeros, then det(A)=0

8. det(A)cannot be 0 to use Cramer's Rule a use of our “useful” number This rule is used to solve a system of equations. Let’s look at two equations and two variables first. D is the determinant of the coefficient matrix. Dx is the determinant of the coefficient matrix with the x column replaced by the constants. Dy is the determinant of the coefficient matrix with the y column replaced by constants. D is the determinant of the coefficient matrix.Dx is the determinant of the coefficient matrix with the x column replaced by the constants. Dy is the determinant of the coefficient matrix with the y column replaced by constants.

9. 24 12 168 42 42 -84 so x = 4, y = -2 D is the determinant of the coefficient matrix. Dx is the determinant of the coefficient matrix with the x column replaced by the constants.Dy is the determinant of the coefficient matrix with the y column replaced by constants. D is the determinant of the coefficient matrix. Dx is the determinant of the coefficient matrix with the x column replaced by the constants. Dy is the determinant of the coefficient matrix with the y column replaced by constants. D is the determinant of the coefficient matrix. Dx is the determinant of the coefficient matrix with the x column replaced by the constants.Dy is the determinant of the coefficient matrix with the y column replaced by constants.

10. Cramer's Rule For 3 equations and 3 variables 2 - 2 1 - 1 - 1 - 1 = -1 = 2 = - 2 = 1 x = -2, y = 2, z = -1 (- 2, 2, -1)