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Chapter 8: Problem 59

Explain how to find the multiplicative inverse for a \(3 \times 3\) invertible matrix.

Short Answer

Expert verified
To find the multiplicative inverse for a \(3 \times 3\) invertible matrix, ensure the matrix is invertible (i.e. the determinant is non-zero), find the matrix's adjoint, and then, apply the formula for the inverse of a matrix: \[ A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A) \]

Step by step solution

01

Verify if the Matrix is Invertible

The first step is to ascertain if the matrix is invertible. A matrix is invertible if and only if the determinant is non-zero. So, compute the determinant of the matrix and ensure it's not zero.
02

Compute Adjoint Matrix

Calculate the adjoint of the matrix. This is done by finding the matrix of minors, applying a checkerboard of negatives to obtain the matrix of cofactors, and transposing the matrix of cofactors.
03

Compute Inverse Matrix

Compute the inverse of the matrix using the formula: \[ A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A) \] where \( A^{-1} \) is the inverse of matrix A, det(A) is the determinant of A, and adj(A) is the adjoint of A.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Determinant Calculation
The determinant is a special number that can be calculated from a square matrix. It is crucial in determining whether a matrix is invertible. For a 3x3 matrix, the determinant is calculated using the formula:\[\text{det}(A) = a(ei 鈭� fh) 鈭� b(di 鈭� fg) + c(dh 鈭� eg)\]where the elements of the matrix are labeled as:
  • Top row: \(a, b, c\)
  • Middle row: \(d, e, f\)
  • Bottom row: \(g, h, i\)
When you calculate the determinant and find that it is not zero, the matrix is invertible. If the determinant equals zero, it is not possible to find an inverse for that matrix. This number helps you determine the next steps, like calculating the inverse.
Adjoint Matrix
Before finding the inverse, you need to compute the adjoint matrix (also known as adjugate). The adjoint is the transpose of the cofactor matrix. Each element of the original matrix has a minor, and these minors form a new matrix called the matrix of minors. Following that, we apply signs in a checkerboard pattern to find the cofactor matrix, then transpose it to get the adjoint matrix. By rearranging the elements, the matrix becomes suitable for the inverse calculation. Transposition is simply switching rows with columns.
Matrix of Minors
The matrix of minors is a key step in the process of finding the adjoint. For each element in the original matrix, we form a smaller 2x2 matrix by removing the row and column containing that element. The determinant of each of these 2x2 matrices is the minor of the corresponding element in the original matrix. By calculating the minor for each element, you create a new matrix, which is then used to find the cofactor matrix. This new matrix is just as large as the original, maintaining the dimensions but changing each number to reflect their corresponding determinant of the 2x2 matrix.
Cofactors
The matrix of cofactors is constructed by adding a sign to each minor. This sign is determined by the position of the element; it's described as a checkerboard pattern of positives and negatives starting with a positive sign (+) for the top-left element. The matrix of cofactors can be expressed as:
  • Position (row 1, column 1): sign is +
  • Position (row 1, column 2): sign is -
  • Position (row 1, column 3): sign is +
  • And so on in an alternating pattern
This change effectively prepares the matrix for the transpose operation to form the adjoint, which is later used in calculating the inverse.
Transpose of a Matrix
The transpose of a matrix involves swapping its rows with its columns. When dealing with matrices, visualizing this is simple; imagine writing each row as a column. The process is necessary when forming the adjoint matrix from the cofactor matrix. Thus, every element \(a_{ij}\) in the original cofactor matrix becomes \(a_{ji}\) in the transpose. This transformation ensures that the cofactor matrix is turned into the adjoint suitable for use in finding the inverse. The transposition, while a straightforward step, is critical as it ensures that your final calculations for inversions are correctly aligned.

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Most popular questions from this chapter

Describe when the multiplication of two matrices is not defined.

Solve: \(\log (x+4)-\log (x-2)=\log x\) (Section \(3.4,\) Example 8 )

a. Write each linear system as a matrix equation in the form \(A X=B\) b. Solve the system using the inverse that is given for the coefficient matrix. $$\left\\{\begin{aligned}w-x+2 y &=-3 \\\x-y+z &=4 \\\\-w+x-y+2 z &=2 \\ -x+y-2 z &=-4\end{aligned}\right.$$ The inverse of \(\left[\begin{array}{rrrr}1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2\end{array}\right]\) is \(\left[\begin{array}{rrrr}0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1\end{array}\right]\)

Will help you prepare for the material covered in the next section. Multiply and write the linear system represented by the following matrix multiplication: $$ \left[\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right]\left[\begin{array}{l} x \\\y \\\z\end{array}\right]=\left[\begin{array}{l}d_{1} \\\d_{2} \\\d_{3}\end{array}\right]$$

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\). $$\left\\{\begin{array}{rr}x-y+z= & -6 \\\4 x+2 y+z= & 9 \\\4 x-2 y+z= & -3\end{array}\right.$$
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