Problem 6 - Differential Equations and Linear Algebra

Question

If $$ A=\left[\begin{array}{rrrr} -2 & 9 & 0 & -1 \\ 4 & -6 & 8 & 8 \\ 0 & -1 & -3 & 4 \\ 7 & -7 & 3 & 1 \end{array}\right] $$ determine the minors \(M_{41}, M_{22}, M_{23}, M_{43},\) and the corresponding cofactors.

Step-by-Step Solution

Answer

The minors and corresponding cofactors for the given matrix are: \(M_{41} = 478, C_{41} = -478\); \(M_{22} = 177, C_{22} = 177\); \(M_{23} = -145, C_{23} = 145\); and \(M_{43} = -104, C_{43} = 104\).

Step 1: Find M_{41}#

Delete row 4 and column 1 from A: $$ A^{'}=\left[\begin{array}{rrr} 9 & 0 & -1 \\ -6 & 8 & 8 \\ -1 & -3 & 4 \end{array}\right] $$ Evaluate the determinant of the obtained matrix (we can use the first row for this): $$ M_{41} = 9\begin{vmatrix} 8&8\\ -3&4 \end{vmatrix} -0\begin{vmatrix} -6&8\\ -1&4 \end{vmatrix} -(-1)\begin{vmatrix} -6&8\\ -1&-3 \end{vmatrix} $$ Now, computing each of those determinants: $$ M_{41} = 9(8\cdot4 - (-3)\cdot8) - 0(0) - 1(-6\cdot(-3) - 8\cdot(-1)) = 9\cdot56 - 0 - 26 = 478 $$ So, \(M_{41} = 478\).

Step 2: Find M_{22}#

Delete row 2 and column 2 from A: $$ A^{'}=\left[\begin{array}{rrr} -2 & 0 & -1 \\ 0 & -3 & 4 \\ 7 & 3 & 1 \end{array}\right] $$ Evaluate the determinant of the obtained matrix (we can use the second column for this): $$ M_{22} = -2\begin{vmatrix} -3&4\\ 3&1 \end{vmatrix} -0(0) +7\begin{vmatrix} 0&-3\\ 7&3 \end{vmatrix} $$ Now, computing each of those determinants: $$ M_{22} =(-2)((-3)\cdot1 - 4\cdot3) + 0 + 7(0\cdot3 - (-3)\cdot7)=-2\cdot(-15)+0+147=30+147=177 $$ So, \(M_{22} = 177\).

Step 3: Find M_{23}#

Delete row 2 and column 3 from A: $$ A^{'}=\left[\begin{array}{rrr} -2 & 9 & -1 \\ 0 & -1 & 4 \\ 7 & -7 & 1 \end{array}\right] $$ Evaluate the determinant of the obtained matrix (we can use the first row for this): $$ M_{23} = (-2)\begin{vmatrix} -1&4\\ -7&1 \end{vmatrix} - 9\begin{vmatrix} 0&4\\ 7&1 \end{vmatrix}-(-1)\begin{vmatrix} 0&-1\\ 7&-7 \end{vmatrix} $$ Now, computing each of those determinants: $$ M_{23}= (-2)((-1)\cdot1-4\cdot(-7)) - 9(0\cdot1-4\cdot7) - 1(0\cdot(-7)-(-1)\cdot7)=-145 $$ So, \(M_{23} = -145\).

Step 4: Find M_{43}#

Delete row 4 and column 3 from A: $$ A^{'}=\left[\begin{array}{rrr} -2 & 9 & -1 \\ 4 & -6 & 8 \\ 0 & -1 & 4 \end{array}\right] $$ Evaluate the determinant of the obtained matrix (we can use the third column for this): $$ M_{43} = (-1)\begin{vmatrix} -6&8\\ -1&4 \end{vmatrix} - 8\begin{vmatrix} 4&-6\\ 0&-1 \end{vmatrix}+4\begin{vmatrix} 4&-6\\ 0&-1 \end{vmatrix} $$ Now, computing each of those determinants: $$ M_{43} = (-1)((-6)\cdot4 - 8\cdot(-1)) - 8(4\cdot(-1) - 0) + 4(4\cdot(-1) - 0) = -104 $$ So, \(M_{43} = -104\).

Step 5: Find the corresponding cofactors#

Now, we will find the corresponding cofactors using the formula \(C_{ij} = (-1)^{i+j} M_{ij}\): - \(C_{41} = (-1)^{4+1}M_{41}=-478\) - \(C_{22} = (-1)^{2+2}M_{22}=177\) - \(C_{23} = (-1)^{2+3}M_{23}=145\) - \(C_{43} = (-1)^{4+3}M_{43}=104\) The corresponding cofactors are \(C_{41} = -478, C_{22} = 177, C_{23} = 145,\) and \(C_{43} = 104\).

Key Concepts

Determinant Calculation

The determinant is a fundamental property in the study of matrices, and it plays a crucial role in linear algebra. It is a scalar value that can be calculated from the elements of a square matrix. The determinant provides essential information about a matrix, such as whether it is invertible, its eigenvalues, and more.

To calculate the determinant of a 3x3 matrix, say:
\[ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]
we use the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
In essence, this formula involves selecting an element from the matrix, then multiplying it by the determinant of the smaller matrix (called a minor) obtained by deleting the row and column that intersect at that element.

It's important to handle the arithmetic carefully, as a small sign error can lead to an incorrect result. Also, it's essential to remember the position of the element since this affects the sign in the final expression.

Matrix Minors

A matrix minor is a determinant of a smaller matrix, which is formed from a larger matrix by deleting one specific row and one specific column. For example, to find the minor \(M_{ij}\) of an element in matrix \(A\), remove the \(i\)-th row and \(j\)-th column from \(A\).

A minor gives insight into the decomposition of determinants and is instrumental in calculating cofactors and determinants of higher-order matrices. Importantly, the size of the minor is one less in both dimensions compared to the original matrix.

• **Example:** For a 4x4 matrix, a minor would be a 3x3 matrix.

After determining the correct minor matrix, compute its determinant, which is simpler than that of the full matrix. This step is crucial in operations such as cofactor expansion.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method for determining the determinant of a matrix. It involves expanding the determinant across a row or a column using the cofactors of the matrix.

Each cofactor \(C_{ij}\) is calculated as:

• 1. Multiply the minor of an element by \((-1)^{i+j}\).
• 2. The formula for a cofactor is: \( C_{ij} = (-1)^{i+j} M_{ij} \), where \(M_{ij}\) is the minor determinant.

To perform cofactor expansion, choose a row or column that simplifies calculations (often one with zeros). Then use the cofactors in the formula:
\[ \text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + ... + a_{1n}C_{1n} \] for a given row.

• **Tip:** Using a row or column with more zeros makes calculations easier and faster.

By understanding cofactor expansion, we can efficiently calculate the determinant of larger matrices, which is an essential skill in matrix algebra.