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Magazine Chapter 2: Problem 5
Let \\[ A=\left(\begin{array}{lll} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{array}\right) \\] (a) Compute the determinant of \(A\). Is \(A\) nonsingular? (b) Compute adj \(A\) and the product \(A\) adj \(A\).
Short Answer
Expert verified
(a) The determinant of A is -4, so A is nonsingular.
(b) The adjugate of A is
\\[
\left(\begin{array}{ccc}
-1 & -2 & -2 \\
1 & -3 & 2 \\
-1 & 2 & -1
\end{array}\right)
\\]
and the product A adj A is
\\[
\left(\begin{array}{ccc}
0 & 4 & -4 \\
0 & 8 & -8 \\
0 & 12 & -12
\end{array}\right)
\\]
Step by step solution
01
Compute the determinant of matrix A
Determinant can be calculated using cofactor expansion. Here, we will calculate the determinant using the first row:
\\[
\text{det}(A) = 1 \cdot \left|\begin{array}{cc} 3&4 \\ 4&5 \end{array}\right| - 2 \cdot \left|\begin{array}{cc} 2&4 \\ 3&5 \end{array}\right| + 3 \cdot \left|\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right|
\\]
Now, compute the determinants of the matrices on the right-hand side:
\\[
\text{det}(A) = 1(3 \cdot 5 - 4 \cdot 4) - 2(2 \cdot 5 - 3 \cdot 4) + 3(2 \cdot 4 - 3 \cdot 3)
\\]
Using this, we can evaluate the determinant of A.
02
Determine if A is nonsingular
A matrix is nonsingular if its determinant is nonzero. Compute the determinant of A from Step 1:
\\[
\text{det}(A) = 1(15-16) - 2(10-12) + 3(8-9) = -1 -(-4) + (-3) = -4
\\]
Since the determinant of A is not equal to zero (-4 鈮� 0), matrix A is nonsingular.
03
Compute adjugate of A (adj A)
The adjugate of a matrix is the transpose of the cofactor matrix. Compute the cofactor matrix, and then take its transpose.
First, find the cofactors of each element in A:
\\[
C=\left(\begin{array}{lll}
C_{11} & C_{12} & C_{13} \\
C_{21} & C_{22} & C_{23} \\
C_{31} & C_{32} & C_{33}
\end{array}\right) =
\left(\begin{array}{lll}
(15 - 16) & -(8 - 9) & (5 - 6) \\
-(10 - 12) & (6 - 9) & -(4 - 6) \\
(6 - 8) & -(4 - 6) & (2 - 3)
\end{array}\right) =
\left(\begin{array}{lll}
-1 & 1 & -1 \\
-2 & -3 & 2 \\
-2 & 2 & -1
\end{array}\right)
\\]
Now, transpose the cofactor matrix to find the adjugate of A:
\\[
\text{adj}(A) = C^T =
\left(\begin{array}{lll}
-1 & -2 & -2 \\
1 & -3 & 2 \\
-1 & 2 & -1
\end{array}\right)
\\]
04
Compute the product A adj A
Multiply matrix A and adj A using matrix multiplication rules:
\\[
A \cdot \text{adj}(A) = \left(\begin{array}{ccc}
1 & 2 & 3 \\
2 & 3 & 4 \\
3 & 4 & 5
\end{array}\right) \cdot
\left(\begin{array}{ccc}
-1 & -2 & -2 \\
1 & -3 & 2 \\
-1 & 2 & -1
\end{array}\right)
\\]
Resulting matrix is:
\\[
A \cdot \text{adj}(A) = \left(\begin{array}{ccc}
0 & 4 & -4 \\
0 & 8 & -8 \\
0 & 12 & -12
\end{array}\right)
\\]
Now, the output consists of the determinant of A, whether A is nonsingular, adj A, and the product A adj A:
(a) The determinant of A is -4, so A is nonsingular.
(b) The adjugate of A is
\\[
\left(\begin{array}{ccc}
-1 & -2 & -2 \\
1 & -3 & 2 \\
-1 & 2 & -1
\end{array}\right)
\\]
and the product A adj A is
\\[
\left(\begin{array}{ccc}
0 & 4 & -4 \\
0 & 8 & -8 \\
0 & 12 & -12
\end{array}\right)
\\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
nonsingular matrix
When discussing matrices, calling a matrix "nonsingular" is the same as saying it is "invertible." The key characteristic of a nonsingular matrix is that its determinant is not zero. This trait means the matrix has a unique solution when used in a system of linear equations. For example, if you have a matrix \( A \) and the calculated determinant is \( -4 \), since \( -4 eq 0 \), matrix \( A \) is nonsingular.
- A nonsingular matrix provides a lot of properties. It guarantees that for each input there is a unique output, simplifying mathematical problem-solving in matrix applications.
- In contrast, a matrix with a determinant equal to zero is called "singular," and it is not invertible.
- Checking if a matrix is nonsingular is essential in various fields, including computer graphics, statistics, and engineering.
adjugate of a matrix
The adjugate of a matrix, often referred to as the "adjoint," is crucial in finding the inverse of a matrix. To compute it, take the transpose of the cofactor matrix of the original matrix. This means swapping the rows and columns of the cofactor matrix.
- The adjugate matrix comes into play when you need to find the inverse of a square matrix. The inverse is given by the formula \( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \).
- For the matrix \( A \), first determine the cofactor of each element 鈥� these are the determinants of submatrices formed by removing the row and column of the current element, adjusted with a sign based on their position.
- Once the cofactor matrix is formed, transpose it to obtain the adjugate matrix \( \text{adj}(A) \).
cofactor expansion
Cofactor expansion, sometimes known as Laplace's Expansion, is a method used to calculate the determinant of a matrix. It involves breaking down a matrix into smaller parts and is especially useful for quickly calculating determinants.
- For any matrix, you can choose any row or column from which to expand. However, choosing a row or column with more zeros can often simplify calculations.
- In cofactor expansion, each element in the chosen row (or column) is multiplied by its corresponding cofactor.
- A cofactor is calculated as the determinant of a smaller matrix formed by eliminating the row and column of the element, then adjusted with a sign according to its position: \((-1)^{i+j}\)
- This process results in a sum of calculated products, yielding the determinant of the full matrix.
matrix multiplication
Matrix multiplication is a fundamental operation in linear algebra. It involves taking two matrices and producing a new matrix. This process is not similar to multiplying individual numbers, and it requires careful attention to the dimensions and arrangement of matrices.
- When multiplying matrices \( A \) and \( B \), the number of columns in \( A \) must equal the number of rows in \( B \). The resulting matrix will have the dimensions of the number of rows of \( A \) and the number of columns of \( B \).
- To find each element in the resulting matrix, take the dot product of the appropriate row from the first matrix and the column from the second matrix.
- Matrix multiplication is not commutative, which means \( AB eq BA \) in most cases.
- This operation is crucial for transforming and moving data, and it is widely used in calculations involving linear transformations and more complex algorithms.
- Understanding matrix multiplication is vital in applications from graphics to machine learning, as it helps model complex systems and perform varied computations efficiently.
