Introduction
The determinant of the matrix is a number that is determined for a square matrix. A determinant is useful to find the solution and do the analysis of a system of linear equations. Determinants have many applications in engineering, science, social sciences, and economics. A determinant is a scalar quantity and possesses properties of the linear transformation of a matrix. The determinant of a matrix A is represented by det(A), or |A|.
The determinant is a function from a set of square matrices to a set of real numbers, and it satisfies three properties as given in the following:
| I |=1.
A determinant is linear in rows of a matrix.
If two rows of matrix M are equal, then | M |=0
The second condition is the most important. It means that any rows of the matrix are written as a linear combination of two other vectors, and the determinant is calculated by splitting that row.
Important Properties of Determinants
1. Reflection Property:
The determinant value remains the same if its rows and columns are interchanged. This is called the property of reflection.
2. All-zero Property:
The determinant is zero if all elements of a row or column are zero.
3. Proportionality (Repetition) Property:
If all row or column elements are identical to some other row or column elements, then its determinant is zero.
4. Switching Property:
The sign of a determinant changes if its rows and columns are interchanged.
5. Scalar Multiple Property:
If all the elements of a row or column of a determinant are multiplied by a non-zero value, then the determinant is multiplied by the same value.
6. Sum Property:
The determinant is expressed as the sum of two or more determinants if elements of a row or column of a determinant can be expressed as the sum of two or more terms.
Matrix Determinant Worksheet
Determinants of 2x2 matrices worksheet
1. Find the determinant of the following matrix :
\[A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\]
2. Find the determinant of the matrix given below.
\[B = \begin{bmatrix}-5 & -4 \\ -2 & -3\end{bmatrix}\]
Determinants of 2x2 matrices worksheet answers
1. Find the determinant of the following matrix.
\[A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\]
This is an example of a matrix where all elements are positive.
\[det = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\left ( 1 \right )\left ( 4 \right )-\left ( 2 \right )\left ( 3 \right )\]
= 4- 6
= -2
2. Find the determinant of the matrix below.
\[B = \begin{bmatrix}-5 & -4 \\ -2 & -3\end{bmatrix}\]
This is an example of a matrix where all elements are negative. The basic rules should always be applied when multiplying integers. The product of values with the same sign will always be positive and if the signs are different, the product will always be negative.
\[det = \begin{bmatrix}-5 & -4 \\ -2 & -3\end{bmatrix}\left ( -5 \right )\left ( -3 \right )-\left ( -4 \right )\left ( -2 \right )\]
= 15 - 8
= 7
Few Important points on 3x 3 Determinant Matrix:
The scalar multipliers of a 2 x 2 matrix have top row elements a, b, and c serving to it.
The scalar value is multiplied by 2 x 2 matrix of the remaining elements when the horizontal and vertical line segments pass through a.
Hence we construct the 2 x 2 matrix for scalar multipliers b and c.
The determinant of 3 x 3 matrix is given by,
(Image will be uploaded soon)
(Image will be uploaded soon)
1. Calculate the determinant of the following 3x3 matrix.
\[\begin{vmatrix}2 & -3 & 1\\ 2 & 0 & -1\\ 1 & 4 & 5\end{vmatrix}\]
Solution:
Use 3 x 3 determinant formula:
Applying the formula,
(Image will be uploaded soon)
= 2( 0 – (-4)) + 3 (10 – (-1)) +1 (8-0)
= 2 (0+4) +3 (10 +1) + 1(8)
= 2(4) +3(11) + 8
= 8+33+8
= 49
Therefore, the determinant = 49
2. Calculate the determinant of the 3 x 3 matrix.
\[\begin{vmatrix}1 & 3 & 2\\ -3 & -1 & -3\\ 2 & 3 & 1\end{vmatrix}\]
Solution:
Use the 3 x 3 determinant formula:
(Image will be uploaded soon)
= 1( -1 – (-9)) – 3 (-3 – (-6)) + 2 (-9 – (-2))
= 1 (-1+9) -3 (-3 +6) + 2(-9 + 2)
= 1(8) -3(3) +2(-7)
= 8 -9-14
= -15
Therefore, the determinant =-15
Exercise 1
Prove that the following determinants are zero:
\[A=\begin{vmatrix}1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b \end{vmatrix}\]
\[B=\begin{vmatrix}a & 3a & 4a\\ a & 5a & 6a\\ a & 7a & 8a\end{vmatrix}\]
Given that |A|=5, calculate the value of the other determinants.
\[A=\begin{vmatrix}x & y & z \\ 3 & 0 & 2 \\ 1 & 1 & 1 \end{vmatrix}\]
\[B=\begin{vmatrix}2x & 2y &2z \\ \frac{3}{2} & 0 &1 \\ 1 & 1 & 1\end{vmatrix}\]
\[C=\begin{vmatrix}x & y & z\\ 3x+3 & 3y & 3z+2\\ x+1 & y+1 & z+1\end{vmatrix}\]
Exercise 2
Prove that the determinant is divisible by 21:
\[A=\begin{vmatrix}1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 5 & 6 & 1 \\ 3 & 4 & 5 & 6 & 1 & 2 \\ 4 & 5 & 6 & 1 & 2 & 3 \\ 5 & 6 & 1 & 2 & 3 & 4 \\ 6 & 1 & 2 & 3 & 4 & 5 \end{vmatrix}=21\],
Determinant of a matrix worksheet
Solution of exercise 1
Prove that the following determinants are zero:
\[A=\begin{vmatrix}1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b\end{vmatrix}\]
\[=\begin{vmatrix}1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c\end{vmatrix}\]
\[=(a+b+c)\begin{vmatrix}1 & a & 1\\ 1 & b & 1\\ 1 & c & 1\end{vmatrix}=0\]
\[B=\begin{vmatrix}a & 3a & 4a\\ a & 5a & 6a\\ a & 7a & 8a\end{vmatrix}=0\]
It has two proportional lines. The third column of the matrix equals the sum of the first two columns.
Solution of exercise 2
Prove that the given determinant is divisible by 21:
\[A=\begin{vmatrix}1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 5 & 6 & 1 \\ 3 & 4 & 5 & 6 & 1 & 2 \\ 4 & 5 & 6 & 1 & 2 & 3 \\ 5 & 6 & 1 & 2 & 3 & 4 \\ 6 & 1 & 2 & 3 & 4 & 5 \end{vmatrix}=21\]
\[c_{6}=c_{1}+c_{2}+c_{3}+c_{4}+c_{5}+c_{6}\]
\[=\begin{vmatrix}1 & 2 & 3 & 4 & 5 & 21 \\ 2 & 3 & 4 & 5 & 6 & 21 \\ 3 & 4 & 5 & 6 & 1 & 21 \\ 4 & 5 & 6 & 1 & 2 & 21 \\ 5 & 6 & 1 & 2 & 3 & 21 \\ 6 & 1 & 2 & 3 & 4 & 21 \end{vmatrix}\]
\[21.\begin{vmatrix}1 & 2 & 3 & 4 & 5 & 1 \\ 2 & 3 & 4 & 5 & 6 & 1 \\ 3 & 4 & 5 & 6 & 1 & 1 \\ 4 & 5 & 6 & 1 & 2 & 1 \\ 5 & 6 & 1 & 2 & 3 & 1 \\ 6 & 1 & 2 & 3 & 4 & 1 \end{vmatrix}=21\]
