Question

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rrr} 4 & 0 & 2 \\ -3 & 2 & 1 \\ 1 & -1 & 1 \end{array}\right] $$

Short answer

The minors and cofactors of a matrix are calculated by first obtaining the 2x2 determinant for each element of the matrix, followed by multiplication with \((-1)^{i+j}\) to get the corresponding cofactors. The resulting minor and cofactor matrices are the solution to this exercise.

Step-by-step solution

Compute Minors

Calculate the minor of each element of the matrix. Here, each element is denoted by \(a_{ij}\) where \(i\) represents rows and \(j\) represents columns. For example, for element \(a_{11} = 4\), remove the first row and first column, so the minor \(M_{11}\) is the determinant of the matrix \[ \begin{matrix} 2 & 1 \ -1 & 1 \end{matrix} \]. Carry out this process for all the elements of the 3x3 matrix.

Determine the cofactor matrix

Having found all the minors, calculate the corresponding cofactors. The cofactor of an element \(a_{ij}\) is given by \(C_{ij} = (-1)^{i+j}M_{ij}\), where \(M_{ij}\) is the minor of \(a_{ij}\). For instance, the cofactor of \(a_{11}\) would be \(C_{11} = (-1)^{1+1}M_{11}\). Do this for each element of the 3x3 matrix.

Compile results

The result includes the minor for every element (matrix of minors) as well as the cofactor for all elements (matrix of cofactors).

Key concepts

Minors of a Matrix

In the context of matrix algebra, a "minor" refers to a simplified form of a determinant that's specific to one element of a larger matrix. Imagine a matrix like the one from the exercise, and let's talk about how to find a minor for, say, the element in the first row and first column, denoted as \( a_{11} \). The process is simple:

• Remove the row and column in which the element is located.
• This leaves you with a smaller matrix.
• The determinant of this smaller matrix is the "minor" of that element.

For instance, to find the minor of the element \( a_{11} = 4 \), you would write down the remaining 2x2 matrix:\[\begin{bmatrix} 2 & 1 \ -1 & 1 \end{bmatrix}\]Then, the determinant of this matrix is calculated as \( (2)(1) - (-1)(1) = 2 + 1 = 3 \), so the minor \( M_{11} = 3 \).
Remember, every element will have a corresponding minor, and you follow this process for each one.

Cofactors of a Matrix

A "cofactor" of an element in a matrix is closely related to the concept of minors, with one additional step that involves applying a sign change. Once you've found the minor for an element, obtaining the cofactor involves:

• Calculating the minor of that element.
• Multiplying the minor by \((-1)^{i+j}\), where \(i\) is the row number and \(j\) is the column number of the element.

Let's stick with our previous example of the element \(a_{11} = 4\). We already calculated its minor as \(M_{11} = 3\). To find the cofactor:
The position is the first row and first column, thus \((i+j) = 1+1 = 2\).
Compute the cofactor: \(C_{11} = (-1)^{2}\times 3 = 1 \times 3 = 3\).
The same technique applies to all matrix elements. This signed minor transformation is crucial when determining matrix determinants or performing more advanced operations like matrix inversion.

Determinants

Determinants are a fundamental component of matrix algebra, playing an essential role in systems of equations, transformations, and more. A determinant is a special number that can be calculated from the elements of a square matrix. For a 3x3 matrix, like the one in the exercise, the determinant is calculated using the formula:\[\text{det}(A) = a_{11}(C_{11}) + a_{12}(C_{12}) + a_{13}(C_{13})\]where each \(a_{ij}\) is an element of the matrix and \(C_{ij}\) is its corresponding cofactor. To compute the determinant for our example matrix:
Plug in the values for \(a_{11}\), \(a_{12}\), and \(a_{13}\) and their cofactors. This sums up to a single numerical value representing the matrix. The sign flips introduced by each cofactor ensure the determinant accurately provides insights into the matrix's properties, such as invertibility:

• If the determinant is zero, the matrix does not have an inverse, making it singular.
• A non-zero determinant means the matrix is invertible.

Understanding how to calculate determinants gives insights into the behavior and characteristics of matrices.