3.9 Determinants det(A) or |A| a b a b det = = ad - bc c d c d Given a square matrix A its determinant is a real number associated with the matrix. The determinant of A is written: det(A) or |A| For a 2x2 matrix, the definition is a b a b det = = ad - bc c d c d For larger matrices the definition is more complicated

Determinants 2x2 examples 1 3 2 4 1 3 2 4 det = = (1)(4) – (2)(3) = -2 -5 -2 2 det = = (-5)(0) – (2)(-2) = 4 1 2 4 det = = (1)(4) – (2)(2) = 0

Determinants M11 : remove row 1, col 1 -2 1 -1 2 7 3 A = 2 3 M11 = 7 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M11 : remove row 1, col 1 -2 1 -1 2 7 3 A = 2 3 M11 = 7

Determinants M12 : remove row 1, col 2 -2 1 -1 2 7 3 A = -1 3 M12 = 2 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M12 : remove row 1, col 2 -2 1 -1 2 7 3 A = -1 3 M12 = 2

Determinants M13 : remove row 1, col 3 -2 1 -1 2 7 3 A = -1 2 M13 = 2 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M13 : remove row 1, col 3 -2 1 -1 2 7 3 A = -1 2 M13 = 2 7

Determinants M21 : remove row 2, col 1 1 -1 2 7 3 A = -2 1 -2 M21 = 7 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M21 : remove row 2, col 1 1 -1 2 7 3 A = -2 1 -2 M21 = 7

Determinants M22 : remove row 2, col 2 1 -1 2 7 3 A = -2 1 -2 M22 = 2 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M22 : remove row 2, col 2 1 -1 2 7 3 A = -2 1 -2 M22 = 2

Determinants M23 : remove row 2, col 3 1 -1 2 7 3 A = -2 1 1 M23 = 2 7 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M23 : remove row 2, col 3 1 -1 2 7 3 A = -2 1 1 M23 = 2 7

Determinants M31 : remove row 3, col 1 1 -1 2 7 3 A = -2 1 -2 M31 = 2 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M31 : remove row 3, col 1 1 -1 2 7 3 A = -2 1 -2 M31 = 2 3

Determinants M32 : remove row 3, col 2 1 -1 2 7 3 A = -2 1 -2 M32 = -1 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M32 : remove row 3, col 2 1 -1 2 7 3 A = -2 1 -2 M32 = -1 3

Determinants M33 : remove row 3, col 3 1 -1 2 7 3 A = -2 1 1 M33 = -1 To define det(A) for larger matrices, we will need the definition of a minor Mij The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M33 : remove row 3, col 3 1 -1 2 7 3 A = -2 1 1 M33 = -1 2

3.9.1 The formula for a 3x3 matrix For a matrix a11 A = a21 a12 a31 a22 a32 a13 a23 a33 Its determinant is given by |A| = a11|M11| - a12|M12| + a13|M13| From the formula for a 2x2 matrix: |M11|= = a22a33 - a23a32 a22 a32 a23 a33

3.9.1 The formula for a 3x3 matrix For a matrix a11 A = a21 a12 a31 a22 a32 a13 a23 a33 Its determinant is given by |A| = a11|M11| - a12|M12| + a13|M13| From the formula for a 2x2 matrix: a21 a23 |M12|= = a21a33 - a23a31 a31 a33

3.9.1 The formula for a 3x3 matrix For a matrix a11 A = a21 a12 a31 a22 a32 a13 a23 a33 Its determinant is given by |A| = a11|M11| - a12|M12| + a13|M13| From the formula for a 2x2 matrix: a21 a22 |M13|= = a21a32 - a31a22 a31 a32

3x3 Example -2 1 -1 2 7 3 A = |A| = 1x|M11| - 1x|M12| + (-2)x|M13| A = |A| = 1x|M11| - 1x|M12| + (-2)x|M13| |A|= 1x - 1x + (-2) 2 7 3 -1 = 1x(-21) -1x(-6) +(-2)x(-11) = 7

3x3 Example 3 1 5 -1 2 B = |B| = 0x|M11| - 1x|M12| + 3x|M13| 1 5 -1 2 B = |B| = 0x|M11| - 1x|M12| + 3x|M13| |B|= 0x - 1x + 3 x 3 2 1 5 -1 = 0x(-2) -1x(1) +(3)x(13) = 38

3.9.1 The formula for a 3x3 matrix For the matrix a11 A = a21 a12 a31 a22 a32 a13 a23 a33 We used the top row to calculate the determinant: |A| = a11|M11| - a12|M12| + a13|M13| However, we could equally have used any row of the matrix and performed a similar calculation

3.9.1 The formula for a 3x3 matrix For the matrix a11 A = a21 a12 a31 a22 a32 a13 a23 a33 Using the top row: |A| = a11|M11| - a12|M12| + a13|M13| Using the second row |A| = -a21|M21| + a22|M22| - a23|M23| Using the third row |A| = a31|M31| - a32|M32| + a33|M33|

3.9.1 The formula for a 3x3 matrix |A| = a11|M11| - a12|M12| + a13|M13| = -a21|M21| + a22|M22| - a23|M23| = a31|M31| - a32|M32| + a33|M33| Notice the changing signs depending on what row we use: + -

3.9.1 The formula for a 3x3 matrix Equally, we could have used any column as long as we follow the signs pattern + - a11 A = a21 a12 a31 a22 a32 a13 a23 a33 E.g. using the first column: |A| = a11|M11| - a21|M21| + a31|M31|

This choice sometimes makes it a bit easier to calculate determinants This choice sometimes makes it a bit easier to calculate determinants. e.g. -2 1 2 3 A = Using the first row: |A|= 1x - 1x + (-2) x 2 1 3 = 1x(-1) -1x(0) + (-2)x(0) = -1

This choice sometimes makes it a bit easier to calculate determinants This choice sometimes makes it a bit easier to calculate determinants. e.g. -2 1 2 3 A = |A|= 1x - 0 + 0 = 1x(-1) = -1 2 1 3 However, using the first column:

3.9.2 A general formula for determinants For a 4x4 matrix we add up minors like the 3x3 case, and again use the same signs pattern + - + - - + - + + - + - - + - + Notice that if we think of the signs pattern as a matrix, then it can be written as (-1)i+j

3.9.2 A general formula for determinants For a nxn matrix A=(aij) the co-factors of A are defined by Cij:= (-1)i+j|Mij| The determinant of A is given by the formula |A|= aij Cij for any j=1,2,...,n Or, |A|= aij Cij for any i=1,2,...,n