Problem 35
Question
Evaluate each \(3 \times 3\) determinant. $$\left|\begin{array}{rrr} 1 & 1 & -5 \\ 3 & -7 & -4 \\ 4 & -6 & 9 \end{array}\right|$$
Step-by-Step Solution
Verified Answer
The determinant of the matrix is -180.
1Step 1: Understand the Matrix and the Formula
We have a 3x3 matrix for which we need to calculate the determinant. The formula for finding the determinant of a 3x3 matrix \( A \) with elements \( a_{ij} \) is given as: \[\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]
2Step 2: Identify the Elements
From the matrix:\[\left|\begin{array}{rrr} 1 & 1 & -5 \ 3 & -7 & -4 \ 4 & -6 & 9 \end{array}\right|\]the elements are: \( a_{11} = 1, a_{12} = 1, a_{13} = -5, a_{21} = 3, a_{22} = -7, a_{23} = -4, a_{31} = 4, a_{32} = -6, a_{33} = 9 \).
3Step 3: Apply the Formula
Substitute the identified elements into the determinant formula:\[\det(A) = 1((-7 \times 9) - (-4 \times -6)) - 1((3 \times 9) - (-4 \times 4)) + (-5)((3 \times -6) - (-7 \times 4))\]
4Step 4: Simplify the Determinants
Calculate each of the smaller determinants: 1. \((-7 \times 9) - (-4 \times -6) = -63 - 24 = -87\)2. \((3 \times 9) - (-4 \times 4) = 27 + 16 = 43\)3. \((3 \times -6) - (-7 \times 4) = -18 + 28 = 10\)
5Step 5: Calculate the Overall Determinant
Substitute the simplified values back into the equation:\[\det(A) = 1(-87) - 1(43) + (-5)(10)\]Calculate:\[\det(A) = -87 - 43 - 50 = -180\]
6Step 6: Final Evaluation
The determinant of the matrix is \(-180\). Ensure each calculation step is verified for accuracy to finalize the determinant.
Key Concepts
Understanding Linear AlgebraExploring 3x3 MatricesFundamentals of Determinant CalculationMatrix Operations and Their Importance
Understanding Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It plays a crucial role in various applications such as computer graphics, engineering, and data science. One of the fundamental tasks in linear algebra is to understand the properties and operations of matrices.
The concept of determinants is particularly important, as it helps in analyzing matrix properties like invertibility and the solution of linear systems. By working with determinants, we can understand if a matrix will have a unique solution or if it is singular (no inverse exists).
‘Linear algebra’ often involves performing operations with matrices, determinants being one of them, to solve practical problems efficiently.
The concept of determinants is particularly important, as it helps in analyzing matrix properties like invertibility and the solution of linear systems. By working with determinants, we can understand if a matrix will have a unique solution or if it is singular (no inverse exists).
‘Linear algebra’ often involves performing operations with matrices, determinants being one of them, to solve practical problems efficiently.
Exploring 3x3 Matrices
A 3x3 matrix is an arrangement of numbers in three rows and three columns. Each element in a matrix is denoted by two subscripts, for example, in matrix element \(a_{ij}\), \(i\) represents the row number and \(j\) represents the column number. In our exercise, the matrix is:
Additionally, they are compact enough to handle by hand, but still manage to showcase complex interactions between elements through matrix operations and determinant calculations.
- Row 1: \([1, 1, -5]\)
- Row 2: \([3, -7, -4]\)
- Row 3: \([4, -6, 9]\)
Additionally, they are compact enough to handle by hand, but still manage to showcase complex interactions between elements through matrix operations and determinant calculations.
Fundamentals of Determinant Calculation
Calculating a determinant for a 3x3 matrix can be accomplished by using the rule of Sarrus or expansion by minors.
In our example, we use expansion by minors, which involves breaking down the matrix into smaller parts, called minors. The formula for the determinant of a 3x3 matrix is:
\[ ext{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]
By carefully substituting each element from the matrix into the formula, and calculating the smaller 2x2 determinants, the overall determinant is then the combination of these results.
Determinants provide valuable insights into the matrix, like whether the matrix is invertible or not. They’re an essential part of linear algebra.
In our example, we use expansion by minors, which involves breaking down the matrix into smaller parts, called minors. The formula for the determinant of a 3x3 matrix is:
\[ ext{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]
By carefully substituting each element from the matrix into the formula, and calculating the smaller 2x2 determinants, the overall determinant is then the combination of these results.
Determinants provide valuable insights into the matrix, like whether the matrix is invertible or not. They’re an essential part of linear algebra.
Matrix Operations and Their Importance
Matrix operations are the foundation of linear algebra and involve operations such as addition, subtraction, multiplication, and finding determinants.
These operations allow mathematicians and engineers to perform calculations required in various scientific fields, such as physics and economics.
Determinant calculation falls under matrix operations and helps in determining the types of solutions linear systems might have. If the determinant is zero, the matrix is singular and doesn't have an inverse. During matrix operations, understanding how each element interacts to impact the matrix's determinant is useful.
This knowledge aids in complex calculations, simulations, and mathematical modeling. Learning to perform these operations accurately is a crucial step in mastering linear algebra.
These operations allow mathematicians and engineers to perform calculations required in various scientific fields, such as physics and economics.
Determinant calculation falls under matrix operations and helps in determining the types of solutions linear systems might have. If the determinant is zero, the matrix is singular and doesn't have an inverse. During matrix operations, understanding how each element interacts to impact the matrix's determinant is useful.
This knowledge aids in complex calculations, simulations, and mathematical modeling. Learning to perform these operations accurately is a crucial step in mastering linear algebra.
Other exercises in this chapter
Problem 35
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