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Chapter 3: Problem 16

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rrr}-3 & 4 & 2 \\ 6 & 3 & 1 \\ 4 & -7 & -8\end{array}\right]$$

Short Answer

Expert verified
The minors of the matrix are \(\begin{bmatrix}-17 & 27 & -15\4 & -58 & -34\7 & -8 & -2\end{bmatrix}\) and the cofactors are \(\begin{bmatrix}-17 & -27 & -15\-4 & -58 & 34\7 & 8 & -2\end{bmatrix}\)

Step by step solution

01

- Compute Minors

The minors of a matrix are the determinants of the submatrices obtained by removing one row and one column from the original matrix. Let's find the minors for each element of the matrix:鈥� Minor of -3 (\(M_{11}\)) = determinant of the 2x2 matrix obtained by removing the first row and column, i.e \(\begin{bmatrix}3 & 1\-7 & -8\end{bmatrix}\) which equals to (3*(-8) - 1*(-7)) = -17.鈥� Similarly, repeating the steps for each element will give us Minors matrix as follows:\(\begin{bmatrix}-17 & 27 & -15\4 & -58 & -34\7 & -8 & -2\end{bmatrix}\)
02

- Compute Cofactors

Cofactors are calculated by multiplying the minor by (-1) to the power of the sum of the element's row and column number. Let's find the cofactors for each element of the matrix:鈥� Cofactor of -3 (\(C_{11}\)) = (-1)^(1+1) * Minor of -3 = 1 * -17 = -17.鈥� Similarly, repeating the steps for each element will give us Cofactors matrix as follows:\(\begin{bmatrix}-17 & -27 & -15\-4 & -58 & 34\7 & 8 & -2\end{bmatrix}\)

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Key Concepts

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

Understanding Determinants
The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix, such as whether the matrix is invertible (i.e., has an inverse) or not. In simpler terms, the determinant is a value derived from a matrix, giving insights into the geometry it describes.

For a basic 2x2 matrix \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]the determinant is calculated as \(ad - bc\). For larger matrices, like 3x3 matrices, determinants are calculated using a method called cofactor expansion.
  • A zero determinant means the matrix does not have an inverse, indicating the vectors that form the matrix are linearly dependent.
  • While the determinant itself is a singular value, calculating it requires understanding submatrices and cofactors.
Exploring Submatrices
A submatrix is formed by removing one or more rows and/or columns from a larger matrix. This is a crucial step in calculating minors and cofactors. A minor of an element in a matrix is the determinant of a submatrix, created by removing the row and column the element is in.

For example, to find the minor of the element in the first row and first column \( (M_{11}) \) of the matrix:\[ \begin{bmatrix} -3 & 4 & 2 \ 6 & 3 & 1 \ 4 & -7 & -8 \end{bmatrix} \]you would remove the first row and first column to get the submatrix:\[ \begin{bmatrix} 3 & 1 \ -7 & -8 \end{bmatrix} \]Calculating the determinant of this submatrix gives the minor for the element \(-3\).
  • Submatrices are central in determining both minors and cofactors.
  • Each element of the original matrix corresponds to one unique submatrix.
The Role of Cofactor Expansion
Cofactor expansion is a method used to compute the determinant of larger matrices, such as 3x3 or larger. This involves expanding the determinant along a row or a column by multiplying each element by its corresponding cofactor, which is the minor adjusted by a sign based on its position.

The sign adjustment follows the pattern of \((-1)^{i+j}\), where \(i\) is the row number and \(j\) is the column number. This pattern ensures an alternating sign scheme across the matrix.

For instance, the cofactor of element \( (C_{11}) \), located in the first row and first column, is calculated as:\[ (-1)^{1+1} \times \text{Minor of element} \]This ensures the correct sign for the cofactor.
  • Each cofactor is linked directly to its corresponding minor by a simple sign change.
  • The complete determinant of the matrix is found by summing the products of the elements of the row or column and their respective cofactors.

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Most popular questions from this chapter

Prove Theorem 3.9: If \(A\) is a square matrix, then \(\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)\) Getting Started: To prove that the determinants of \(A\) and \(A^{T}\) are equal, you need to show that their cofactor expansions are equal. The cofactors are \(\pm\) determinants of smaller matrices, so you need to use mathematical induction. (i) Initial step for induction: If \(A\) is of order \(1,\) then \(A=\left[a_{11}\right]=A^{T}\) so\(\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)=a_{11}\) (ii) Assume the inductive hypothesis holds for all matrices of order \(n-1 .\) Let \(A\) be a square matrix of order \(n .\) Write an expression for the determinant of \(A\) by expanding in the first row. (iii) Write an expression for the determinant of \(A^{T}\) by expanding in the first column. (iv) Compare the expansions in (ii) and (iii). The entries of the first row of \(A\) are the same as the entries of the first column of \(A^{T} .\) Compare cofactors (these are the \(\pm\) determinants of smaller matrices that are transposes of one another) and use the inductive hypothesis to conclude that they are equal as well.

Let \(A\) be an \(n \times n\) matrix in which the entries of each row sum to zero. Find \(|A|\)

Find the value(s) of \(k\) such that \(A\) is singular. $$A=\left[\begin{array}{lll}0 & k & 1 \\\k & 1 & k \\\1 & k & 0\end{array}\right]$$

Find the volume of the tetrahedron with the given vertices. $$(0,0,0),(0,2,0),(3,0,0),(1,1,4)$$

Use a software program or a graphing utility to find (a) \(|\boldsymbol{A}|\) (b) \(\left|\boldsymbol{A}^{T}\right|,(\mathbf{c})\left|\boldsymbol{A}^{2}\right|,(\mathbf{d})|\boldsymbol{2} \boldsymbol{A}|,\) and \((\mathbf{e})\left|\boldsymbol{A}^{-1}\right|\). $$A=\left[\begin{array}{rrr}3 & 1 & -2 \\\2 & -1 & 3 \\\\-3 & 1 & 2\end{array}\right]$$
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