Determinants by Dr. Shorouk Ossama

The determinant of (2x2) matrix: Where: Then the determinant of A is: -

The determinant of (3x3) matrix: Where: Then the determinant of A is: + - +

Is known as the Minor of a11, (M11) Is known as the Minor of a11, (M11). the minor of an element is obtained by deleting the row and column through the element and writing down the determinant of the elements of the remaining 2 x 2 sub matrix with a + or – sign attached.

The minors of a12 and a13 are: Where the signs attached should be noted.

The signs associated with the factors alternate, starting with a + at the top left as we move across or down from the top left-hand corner as shown: The sign associated with Cij is + if i+j is even and - if i+j is odd, can be expressed as (-1)i+j

So the number (-1)i+j is denoted by Cofactor a13, Row 1 & column 3, so (-1)i+j = (-1)4 = (even) + a21, Row 2 & colunm 1, so (-1)i+j = (-1)3 = (odd) -

Example: Finding Minors and Cofactors for C11 = (-1)1+1 M11 = M11 = 16 so C32 = (-1)2+3 M32 = - M32 = -26

Example: Find the determinant of the matrix = 3 (-4) – [(-2) (-2) - 5 (3)] = -1 + - +

Example: Find the determinant of the matrix = -2 ( 1 + 2) = -6 2 0 0 1 + - +

Properties of Determinant det AT = det A, where AT is the transpose of A. The determinant of a square matrix and its transpose are equal since. Example: Evaluate

= -3 Since the determinant has two zero in the first column, it is advantageous to use Rule 1. the determinant of the transpose of A is given by:

2. If every element of any single row or column of the matrix A is multiplied by a scalar K, then the determinant of this matrix is k det A. = a11 ka22a33 - a11 ka23a32 - a12 ka21a33 + a12 ka23a31 + a13 ka21a32 - a13 ka22a31 = k det A Example: Evaluate Δ = 11 x = 11 x ( -13 - 72 + 1) = -924

3. If B obtained from A by interchanging two rows (or columns) then det B = - det A For example, rows 1 and 3 are interchanged, so that: = a31 a22a13 – a31 a23a12 – a32 a21a13 + a32 a23a11 + a33 a21a12 – a33 a22a11 = - det A The same is true whichever or columns pairs are exchanged.

Example: Evaluate + - + - 4 2 -1 4 4 -1 Interchange rows 1 and 2 using rule 3: Expanding by row 1, remembering the sign role for cofactors: = 2 -1 2 0 -1 + - + - 4 -1 2 0 -1 2 -1 4 4 -1

is an expansion by the second column. 4. Expansion by any row or column By grouping the terms differently, we can write, for example: det A = a31 (a12 a23 – a13 a22) - a32 (a11 a23 – a13 a21) + a33 (a11 a22 – a12 a21) = a31 C31 + a32 C32 + a33 C33 In terms of cofactors. Here the elements a31, a32 ,a33 constitute the third row, and we call this the expansion of det A by the third row. It can be shown that the expansion can be written down similarly using any row or column. Thus: det A = a12 C12 + a22 C22 + a32 C32 is an expansion by the second column.

Example: Evaluate det A, where Since column 3 contains three zeros, expand by this column. The cofactor of the element in rows 3, column 3, is the 3 x 3 determinant obtained from A by deleting the third row and third column in A. it is associated with a + sign. 2 1 0 -1

= 2 (4 + 10) = 28

The Adjoint Matrix: We can now rewrite the formula for the inverse: Example: Find the inverse of: Evaluate first det A = 1 x [-2-1] -2 x [0+1] – 1 x [0-1] = -4 The cofactors are: C11= -3 C12= -1 C13= -1 C21= 5 C22= -1 C23= 3 C31= -1 C32= 1 C33= 1

Exercise: If Find: det A det B det(AB) det AT adjA det A-1

Thanks