Matrix and Determinant - Properties of Determinant
Slide 1
- The determinant is a scalar value that can be calculated for square matrices only.
- It is denoted by |A|.
- For a 2x2 matrix A = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is given by |A| = ad - bc.
Slide 2
Properties of Determinant:
- If A and B are square matrices of the same order, then |A * B| = |A| * |B|.
- Transpose of a matrix does not change the value of its determinant, i.e., |A| = |A^T|.
- If any two rows (or columns) of a matrix are interchanged, the value of its determinant changes its sign.
- If all the elements of a row (or column) in a matrix are multiplied by a constant ‘k’, the value of its determinant is also multiplied by ‘k’.
- If any two rows (or columns) of a matrix are identical, then its determinant is zero.
Slide 3
Properties of Determinant (contd.):
- If a matrix has a row (or column) with all elements as zero, then its determinant is zero.
- If all the elements of a row (or column) in a matrix are zero, then the value of its determinant is zero.
- If a matrix has identical elements along either diagonal, then its determinant is zero.
- If a matrix has a row (or column) with all non-zero elements equal, then the value of its determinant is zero.
- If a row (or column) of a matrix consists of elements which are sum of any two or more matrices, then the determinant of the matrix is the sum of determinants of those matrices.
Slide 4
Example 1:
Consider the matrix A = $\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$. Find |A|.
|A| = 2 * 4 - 1 * 3 = 8 - 3 = 5.
Slide 5
Example 2:
Find the determinant of the matrix B = $\begin{bmatrix} 2 & -1 & 3 \\ 0 & 4 & 2 \\ 1 & 3 & -2 \end{bmatrix}$.
|B| = 2 * [(4 * -2) - (3 * 2)] - (-1) * [(0 * -2) - (1 * 2)] + 3 * [(0 * 3) - (1 * 4)]
= 2 * (-8 - 6) - (-1) * (-4) + 3 * (-4)
= -28 - 4 - 12
= -44.
Slide 6
Example 3:
Calculate the determinant of the matrix C = $\begin{bmatrix} 3 & 1 & 2 \\ 2 & 4 & 0 \\ 1 & -2 & 3 \end{bmatrix}$.
|C| = 3 * [(4 * 3) - (0 * -2)] - 1 * [(2 * 3) - (0 * 1)] + 2 * [(2 * -2) - (1 * 4)]
= 3 * (12 - 0) - 1 * (6 - 0) + 2 * (-4 - 4)
= 3 * 12 - 1 * 6 + 2 * (-8)
= 36 - 6 - 16
= 14.
Slide 7
Example 4:
Find the determinant of the matrix D = $\begin{bmatrix} -1 & 3 \\ 5 & -4 \end{bmatrix}$.
|D| = (-1) * (-4) - 3 * 5 = 4 - 15 = -11.
Slide 8
Example 5:
If |E| = 8, find |2E|.
|2E| = (2 * 2) * |E| = 4 * 8 = 32.
Slide 9
Example 6:
If |F| = -3, find |-F|.
|-F| = -|F| = -(-3) = 3.
Slide 10
Example 7:
If |G| = 0, is it possible for |2G| to be non-zero?
No, because multiplying a matrix with determinant 0 by any scalar will result in a determinant of 0 as well.
Slide 11
Properties of Determinant (contd.):
- The determinant of a null matrix is always zero.
- The determinant of an identity matrix is always one.
- If two rows (or columns) of a matrix are proportional, then its determinant is zero.
- The determinant of a product of two matrices is equal to the product of their determinants, i.e., |A * B| = |A| * |B|.
Slide 12
Properties of Determinant (contd.):
- If A is a square matrix and k is a scalar, then |kA| = k^n * |A|, where n is the order of the matrix A.
- If A is an invertible matrix, then |A^(-1)| = 1/|A|.
- If A and B are square matrices of the same order, then |A + B| may not be equal to |A| + |B|.
- If A and B are similar matrices, then |A| = |B|.
Slide 13
Example 8:
Let A = $\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$ and B = $\begin{bmatrix} -1 & 2 \\ 0 & 3 \end{bmatrix}$. Find |A + B|.
|A + B| = |$\begin{bmatrix} 2 + (-1) & 1 + 2 \\ 3 + 0 & 4 + 3 \end{bmatrix}$| = |$\begin{bmatrix} 1 & 3 \\ 3 & 7 \end{bmatrix}$|
|A + B| = (1 * 7) - (3 * 3) = 7 - 9 = -2.
Slide 14
Example 9:
Find |3B - 2A| if A = $\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}$ and B = $\begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}$.
|3B - 2A| = |3$\begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}$ - 2$\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}$| = |$\begin{bmatrix} 6 & 9 \\ -3 & 12 \end{bmatrix}$ - $\begin{bmatrix} 2 & 8 \\ 0 & -2 \end{bmatrix}$|
= |$\begin{bmatrix} 6 - 2 & 9 - 8 \\ -3 - 0 & 12 - (-2) \end{bmatrix}$|
= |$\begin{bmatrix} 4 & 1 \\ -3 & 14 \end{bmatrix}$|
|3B - 2A| = (4 * 14) - (1 * -3) = 56 + 3 = 59.
Slide 15
Example 10:
Given that |A| = 4, find the determinant of the matrix K = $\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$.
Since K is of the same order as A, |K| = |A| = 4.
Slide 16
Example 11:
Find the determinant of the matrix P = $\begin{bmatrix} -3 & 0 & 1 \\ 5 & 2 & -2 \\ 1 & -1 & 4 \end{bmatrix}$.
|P| = -3 * [(2 * 4) - (-2 * -1)] - 0 * [(5 * 4) - (-2 * 1)] + 1 * [(5 * -1) - (2 * 1)]
|P| = -3 * (8 - 2) - 0 * (20 + 2) + 1 * (-5 - 2)
|P| = -3 * 6 - 0 * 22 + 1 * (-7)
|P| = -18 + 0 - 7
|P| = -25.
Slide 17
Example 12:
If |Q| = 0, find the determinant of K = $\begin{bmatrix} -5 & 2 \\ 6 & k \end{bmatrix}$.
|K| = -5 * k - 2 * 6 = -5k - 12.
Given |Q| = 0,
-5k - 12 = 0
-5k = 12
k = -12/-5
k = 12/5.
Slide 18
Example 13:
Find the determinant of the matrix R = $\begin{bmatrix} k & 3 \\ 2k & 6 \end{bmatrix}$.
|R| = k * 6 - 3 * 2k = 6k - 6k = 0.
Since |R| = 0, we can conclude that the determinant of the matrix R is always zero for any value of k.
Slide 19
Example 14:
Let S be a matrix such that |S| = -40. Find |2S^2|.
|2S^2| = (2^2) * |S^2| = 4 * |-40|
|2S^2| = 4 * (-40) = -160.
Slide 20
Example 15:
If |T| = 5 and |T + U| = 7, find |U|, where U is a square matrix.
|T + U| = 7
|U| = |T + U| - |T|
|U| = 7 - 5
|U| = 2.
Sorry, but I can't continue the text in the current block.