01 · Explore
Introduction to Determinants
A determinant is a unique number associated with every square matrix. It serves as a critical tool in determining whether a system of linear equations has a unique solution.
In previous studies of matrices, we learned that a system of linear equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ can be represented in matrix form. The uniqueness of the solution for such a system depends on the value (a₁b₂ - a₂b₁). This specific value is called the determinant of the coefficient matrix A.
Determinants are only defined for square matrices (matrices where the number of rows equals the number of columns). For a matrix A, the determinant is denoted as |A|, det A, or Δ. It is important to note that |A| represents the 'determinant of A' and not the absolute value or modulus.
The applications of determinants are vast, spanning across Engineering, Science, Economics, and Social Sciences. In this chapter, we focus on determinants of order 1, 2, and 3 with real entries.
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Question
Can a 2x3 matrix have a determinant?
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02 · Explore
Determinants of Order One and Two
The calculation of a determinant depends on the order of the square matrix. For lower orders, the process is straightforward and involves basic arithmetic of the matrix elements.
For a matrix of order 1, such as A = [a], the determinant is simply the value of the element itself. Thus, det(A) = a.
For a matrix of order 2, A = [[a₁₁, a₁₂], [a₂₁, a₂₂]], the determinant is calculated by cross-multiplying the elements. We multiply the elements of the principal diagonal and subtract the product of the other diagonal elements.
The formula for a 2x2 determinant is Δ = a₁₁a₂₂ - a₂₁a₁₂. This result is a scalar value that can be positive, negative, or zero.
Evaluating a 2x2 Determinant
Δ = | 2 4 | / | -1 2 |
Using the formula a₁₁a₂₂ - a₂₁a₁₂: (2)(2) - (-1)(4) = 4 - (-4) = 4 + 4 = 8.
Algebraic Determinant
|(x, x+1), (x-1, x)|
Multiply diagonals: x(x) - (x+1)(x-1). This simplifies to x² - (x² - 1) = x² - x² + 1 = 1.
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Evaluate the determinant of A = [[5, 2], [7, 3]].
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03 · Explore
Determinants of Order Three
A determinant of order 3x3 is evaluated by expanding it into a sum of three determinants of order 2x2.
To find the determinant of a 3x3 matrix, we can expand along any of its three rows or three columns. Regardless of the row or column chosen, the final value remains the same. For efficiency, it is best to expand along the row or column containing the most zeros.
The expansion process involves taking an element, multiplying it by (-1)^(i+j) where i and j are the row and column indices, and then multiplying by the 'minor' determinant formed by deleting the row and column of that element.
The sign convention for a 3x3 determinant follows a checkerboard pattern: [[+, -, +], [-, +, -], [+, -, +]].
Steps for Expansion along Row 1 (R₁)
- 1
Element a₁₁
Multiply a₁₁ by the 2x2 determinant remaining after deleting R₁ and C₁.
- 2
Element a₁₂
Multiply -a₁₂ by the 2x2 determinant remaining after deleting R₁ and C₂.
- 3
Element a₁₃
Multiply a₁₃ by the 2x2 determinant remaining after deleting R₁ and C₃.
- 4
Summation
Add the three results together to get the final determinant value.
The sequential process of reducing a 3x3 determinant into manageable 2x2 parts.
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Why is it recommended to expand along a row with zeros?
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04 · Explore
Area of a Triangle using Determinants
Determinants provide a concise coordinate geometry formula for calculating the area of a triangle.
If a triangle has vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃), its area can be represented as 1/2 times the absolute value of a specific 3x3 determinant. The determinant is constructed by placing the x-coordinates in the first column, y-coordinates in the second, and the number 1 in the third column.
Since area is a physical quantity, it must always be positive. If the determinant calculation yields a negative number, we take its absolute value. If the area is already known and we are solving for a missing coordinate, we must consider both the positive and negative possibilities of the determinant.
A key application of this formula is testing for collinearity. If three points are collinear (lie on the same straight line), they cannot form a triangle, and thus the area calculated by the determinant will be exactly zero.
| Condition | Geometric Interpretation |
|---|---|
| Area > 0 | Points form a valid triangle. |
| Area = 0 | Points are collinear (lie on a line). |
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Find the area of a triangle with vertices (1, 0), (6, 0), and (4, 3).
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05 · Explore
Minors and Cofactors
Minors and cofactors are the building blocks used to define the adjoint and inverse of a matrix.
The Minor (Mᵢⱼ) of an element aᵢⱼ is the determinant obtained by deleting the i-th row and j-th column in which the element lies. If the original determinant is of order n, the minor will be of order n-1.
The Cofactor (Aᵢⱼ) is the minor multiplied by a sign factor. It is defined as Aᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ. This means the cofactor is equal to the minor if the sum of the row and column indices is even, and it is the negative of the minor if the sum is odd.
An important property is that the sum of the products of elements of any row with their corresponding cofactors equals the determinant value. However, if elements of one row are multiplied by the cofactors of a different row, the sum is always zero.
Finding Minors and Cofactors
Δ = |(1, -2), (4, 3)|
M₁₁ = 3, A₁₁ = (-1)²(3) = 3; M₁₂ = 4, A₁₂ = (-1)³(4) = -4; M₂₁ = -2, A₂₁ = (-1)³(-2) = 2; M₂₂ = 1, A₂₂ = (-1)⁴(1) = 1.
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What is the relationship between a minor and a cofactor?
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06 · Explore
Adjoint and Inverse of a Matrix
The adjoint and inverse are essential for solving matrix equations and understanding matrix transformations.
The Adjoint of a square matrix A (denoted as adj A) is the transpose of the matrix of its cofactors. For a 2x2 matrix, the adjoint can be found quickly by interchanging the diagonal elements and changing the signs of the off-diagonal elements.
A square matrix A is 'invertible' if and only if it is non-singular, meaning its determinant |A| is not zero. If |A| = 0, the matrix is called singular and does not have an inverse.
The formula for the inverse of a matrix is A⁻¹ = (1/|A|) * adj A. This inverse is unique for every non-singular matrix. A key property is that A * A⁻¹ = A⁻¹ * A = I, where I is the identity matrix.
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If |A| = 5 and A is of order 3, what is |adj A|?
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07 · Explore
Solving Systems of Linear Equations
Matrix methods provide a systematic way to solve systems of linear equations using the inverse of the coefficient matrix.
A system of linear equations can be written in the form AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants.
If A is non-singular (|A| ≠ 0), the system has a unique solution given by X = A⁻¹B. This is known as the Matrix Method. If the system has at least one solution, it is called 'consistent'; if no solution exists, it is 'inconsistent'.
If |A| = 0, we must check the value of (adj A)B. If (adj A)B is not a zero matrix, the system is inconsistent (no solution). If (adj A)B is a zero matrix, the system may have infinitely many solutions or no solution.
Solving a 2x2 System
2x + 5y = 1; 3x + 2y = 7
A = [[2, 5], [3, 2]], |A| = 4 - 15 = -11. adj A = [[2, -5], [-3, 2]]. A⁻¹ = -1/11 [[2, -5], [-3, 2]]. X = A⁻¹B = -1/11 [[2, -5], [-3, 2]] [1, 7] = -1/11 [-33, 11] = [3, -1]. So x=3, y=-1.
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When is a system of equations considered inconsistent?
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