Key Properties and Applications of Determinants
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,
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(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,
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= 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:
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= 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:
\[A=\begin{vmatrix}1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 5 & 6 & 1 \\ 3 & 4 & 5 & 6 & 1 & 2 \\ 4 & 5 & 6 & 1 & 2 & 3 \\ 5 & 6 & 1 & 2 & 3 & 4 \\ 6 & 1 & 2 & 3 & 4 & 5 \end{vmatrix}=21\]
\[c_{6}=c_{1}+c_{2}+c_{3}+c_{4}+c_{5}+c_{6}\]
\[=\begin{vmatrix}1 & 2 & 3 & 4 & 5 & 21 \\ 2 & 3 & 4 & 5 & 6 & 21 \\ 3 & 4 & 5 & 6 & 1 & 21 \\ 4 & 5 & 6 & 1 & 2 & 21 \\ 5 & 6 & 1 & 2 & 3 & 21 \\ 6 & 1 & 2 & 3 & 4 & 21 \end{vmatrix}\]
\[21.\begin{vmatrix}1 & 2 & 3 & 4 & 5 & 1 \\ 2 & 3 & 4 & 5 & 6 & 1 \\ 3 & 4 & 5 & 6 & 1 & 1 \\ 4 & 5 & 6 & 1 & 2 & 1 \\ 5 & 6 & 1 & 2 & 3 & 1 \\ 6 & 1 & 2 & 3 & 4 & 1 \end{vmatrix}=21\]
FAQs on Determinants Worksheet: Practice & Solutions
1. What is a determinant, and what is its main purpose as per the Class 12 Maths syllabus?
A determinant is a unique scalar value that is associated with a square matrix. Its primary purpose in Class 12 Maths is to serve as a powerful tool in algebra. Key applications include:
- Determining the consistency of a system of linear equations.
- Finding the inverse of a matrix, which is essential for solving linear equations using the matrix method.
- Calculating the area of a triangle when the coordinates of its vertices are given.
2. What is the standard method for calculating the determinant of a 3x3 matrix, a common question in worksheets?
To calculate the determinant of a 3x3 matrix, you can expand along any row or column. The most common method is expanding along the first row:
For a matrix A =
[[a, b, c],
[d, e, f],
[g, h, i]],
the determinant |A| is calculated as:
a(ei - fh) - b(di - fg) + c(dh - eg).
This involves multiplying each element of the first row by the determinant of its corresponding 2x2 minor, with alternating signs (+, -, +).
3. What is the fundamental difference between a matrix and a determinant?
The key difference is that a matrix is a rectangular arrangement of numbers or functions, representing a collection of data or a linear transformation. In contrast, a determinant is a single numerical value calculated from a square matrix only. Think of a matrix as an object (like a system of equations) and its determinant as a specific property of that object (like whether the system has a unique solution).
4. How are determinants applied to find the area of a triangle?
Determinants provide a simple formula to find the area of a triangle whose vertices are (x₁, y₁), (x₂, y₂), and (x₃, y₃). The area is given by the absolute value of:
Area = ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |
This expression is the expansion of the determinant:
Area = ½ |det([[x₁, y₁, 1], [x₂, y₂, 1], [x₃, y₃, 1]])|
An important application is to check for collinearity of three points; if the area (and thus the determinant) is zero, the points lie on a straight line.
5. Why can we only calculate the determinant for a square matrix?
Determinants are fundamentally linked to concepts that only apply to square matrices, such as finding an inverse and solving systems of linear equations with a unique solution (where the number of equations equals the number of variables). A determinant represents the scaling factor of the linear transformation described by a matrix. This concept of volume or area scaling is only well-defined for transformations within the same dimension (e.g., from 2D to 2D), which is what square matrices represent.
6. What does a determinant value of zero signify about a matrix and the system of equations it represents?
A determinant of zero is a critical indicator. It signifies that:
- The matrix is singular, meaning it does not have an inverse.
- The linear transformation represented by the matrix collapses the space into a lower dimension (e.g., a 2D area is mapped to a line or a point).
- The system of linear equations (AX = B) corresponding to the matrix either has no solution or infinitely many solutions. It will not have a unique solution.
7. How do the properties of determinants help in simplifying complex calculations?
The properties of determinants are powerful shortcuts that avoid lengthy expansions. For example, using row or column operations (like Rᵢ → Rᵢ + kRⱼ) does not change the determinant's value. By applying these properties strategically, you can introduce one or more zeros into a row or column. Expanding the determinant along that row or column then becomes much simpler, as many terms become zero, significantly reducing calculation time and the risk of errors.
8. What is the real-world importance of learning to solve determinants beyond passing exams?
Beyond the curriculum, determinants have significant applications in various fields. In engineering and physics, they are used to solve systems of equations for circuit analysis and structural mechanics. In computer graphics, they are essential for 3D modelling and transformations. In economics, they are used in input-output models to analyse economic systems. Learning determinants builds a foundation for these advanced real-world problem-solving techniques.
