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Magazine Chapter 3: Problem 18
Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{ccc}3 & -1 & 0 \\ -3 & 0 & -4 \\ 0 & -1 & -4\end{array}\right]\)
Short Answer
Expert verified
The determinant of the matrix is 0.
Step by step solution
01
Set Up the Determinant
To find the determinant of a 3x3 matrix using cofactor expansion along the first row, we'll use the formula: \ \( \text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \) \ where \( C_{ij} \) is the cofactor of element \( a_{ij} \). Identify the elements of the first row: \( a_{11} = 3 \), \( a_{12} = -1 \), and \( a_{13} = 0 \).
02
Calculate Cofactor for a11
The cofactor \( C_{11} \) is calculated using the submatrix obtained by removing the first row and first column: \ \( \begin{vmatrix} 0 & -4 \ -1 & -4 \end{vmatrix} \). \ The determinant of this 2x2 matrix is \( (0)(-4) - (-4)(-1) = 0 - 4 = -4 \). \ Therefore, \( C_{11} = (-1)^{1+1} \times (-4) = -4 \) (since \((-1)^{1+1} = 1\)).
03
Calculate Cofactor for a12
The cofactor \( C_{12} \) uses the submatrix formed by removing the first row and second column: \ \( \begin{vmatrix} -3 & -4 \ 0 & -4 \end{vmatrix} \). \ The determinant of this matrix is \( (-3)(-4) - (0)(-4) = 12 - 0 = 12 \). \ Thus, \( C_{12} = (-1)^{1+2} \times 12 = -12 \) (since \((-1)^{1+2} = -1\)).
04
Calculate Cofactor for a13
For \( a_{13} \), determine \( C_{13} \) using the submatrix with the first row and third column removed: \ \( \begin{vmatrix} -3 & 0 \ 0 & -1 \end{vmatrix} \). \ The determinant is \( (-3)(-1) - (0)(0) = 3 \). \ Thus, \( C_{13} = (-1)^{1+3} \times 3 = 3 \) (since \((-1)^{1+3} = 1\)).
05
Expand Cofactors
Apply the cofactor expansion formula: \ \( \text{det}(A) = 3 \times (-4) + (-1) \times (-12) + 0 \times 3 \). \ Simplify this: \( = -12 + 12 + 0 \).
06
Calculate Final Determinant
Simplify the expression obtained from the cofactor expansion: \ \( -12 + 12 + 0 = 0 \). \ Therefore, the determinant of the matrix is 0.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cofactor Expansion
Cofactor expansion is a method used to calculate the determinant of a square matrix. It involves breaking down a large matrix into smaller, more manageable submatrices.
This technique is especially beneficial for matrices larger than 2x2, where direct computation becomes cumbersome.
Here's how cofactor expansion works:
This technique is especially beneficial for matrices larger than 2x2, where direct computation becomes cumbersome.
Here's how cofactor expansion works:
- Choose a row or a column to expand along. This exercise selects the first row.
- For each element in the chosen row or column, compute its cofactor.
- The cofactor of an element is the determinant of the smaller matrix that remains after removing the row and column of the element in question, multiplied by \((-1)^{i+j}\) where \(i\) and \(j\) are the element's row and column numbers.
- Finally, multiply each element by its cofactor and sum the results to find the matrix's determinant.
3x3 Matrix
A 3x3 matrix is a matrix with three rows and three columns, making it a square matrix. This specific size of matrix is common in many mathematical applications due to its simplicity yet powerful functionality in linear algebra.
Let's look at a general 3x3 matrix:
Let's look at a general 3x3 matrix:
- The elements in a 3x3 matrix can be represented as \(a_{ij}\) where \(i\) denotes the row number and \(j\) the column number.
- To specifically calculate its determinant, we often use methods like cofactor expansion.
- The determinant of a 3x3 matrix helps in understanding the matrix's properties, such as whether it is invertible.
Matrix Algebra
Matrix algebra is a branch of mathematics that deals with matrices and their operations. Operations include addition, subtraction, multiplication, and finding determinants.
The determinant is a key function in matrix algebra defining crucial properties, such as invertibility.
The determinant is a key function in matrix algebra defining crucial properties, such as invertibility.
- A matrix's determinant is a scalar value that provides insights into the matrix's characteristics.
- In matrix multiplication, understanding determinants is fundamental for working with matrix inverses and solutions to matrix equations.
Linear Algebra
Linear algebra is a field of mathematics concerned with vectors, vector spaces, and linear transformations. It encompasses a wide range of modern mathematical concepts, including matrices.
Matrices are one primary tool in linear algebra, and their determinants are crucial for several reasons:
Matrices are one primary tool in linear algebra, and their determinants are crucial for several reasons:
- Determinants help in determining the solvability of linear systems, indicating whether unique, infinite, or no solutions exist.
- They provide a means of understanding vector space properties, such as spans and eigenvalues.
- Cofactor expansion helps calculate the determinant needed to perform these analyses effectively.
