Question

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

Short answer

The minors of the matrix are: \[\begin{{array}}{{ccc}}20 & -18 & -6\-30 & 0 & 9\-6 & -24 & 12\end{{array}}\]and the cofactors of the matrix are: \[\begin{{array}}{{ccc}}20 & 18 & -6\30 & 0 & -9\-6 & 24 & 12\end{{array}}\]

Step-by-step solution

Calculate the Minors of the Matrix

To find the minor of an element, remove the row and column that the element belongs to, and calculate the determinant of the remaining 2x2 matrix. Do this for all the elements of the matrix.

Calculate the Cofactors of the Matrix

To calculate the cofactor of an element, take the minor of that element and multiply it by \((-1)^{i+j}\), where \(i\) is the row number and \(j\) is the column number. Note that matrix indexing starts from 1. Calculate the cofactor for each element in the matrix.

Key concepts

Matrix Algebra

In the world of mathematics, matrix algebra is a cornerstone for various fields including computer science, engineering, and physics. A matrix is essentially a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Why are they so important? Well, matrices are like a Swiss Army knife for linear equations—they provide a streamlined way to handle multiple equations at once.

For example, you can add and subtract matrices of the same size by adding or subtracting corresponding elements. Multiplication, on the other hand, is a bit trickier. When you multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The result is a new matrix where each element is computed as the sum of products of elements from the rows of the first matrix and the columns of the second one.

To fully engage with matrix algebra, it's necessary to familiarize yourself with notions such as matrix addition, subtraction, scalar multiplication, and matrix multiplication. These operations serve as the building blocks for more complex concepts like the determination of matrix minors, cofactors, and ultimately, the determinant itself.

Determinant of a Matrix

The determinant is a special number associated with square matrices, and it's a powerful tool in linear algebra. It's something like a mathematical 'DNA'—providing vital information about the linear transformation that a matrix represents. Specifically, it can tell us volumes about the invertibility of a matrix, the nature of solutions to systems of linear equations, and even volume distortion during transformations.

Determining the determinant can feel like unlocking a secret code. For a 2x2 matrix, it's a straightforward operation: just multiply the elements on the main diagonal and subtract the product of the off-diagonal elements. However, for larger matrices, the process can become complex. This is where the method of cofactor expansion comes into play. It breaks down the computation of a determinant into smaller, more manageable parts, using the concept of minors and cofactors, which brings us neatly to the next key concept.

Cofactor Expansion

Once you're dealing with a square matrix larger than 2x2, finding the determinant through direct calculations becomes cumbersome. This is where cofactor expansion shines—it's a systematic way of breaking down a big problem into smaller ones. To understand cofactor expansion (also known as Laplace's expansion), you should first get to know two terms: minors and cofactors.

A minor is the determinant of a smaller matrix that you get by removing one row and one column from the original matrix—essentially, it's like looking at the matrix through a cut-out that blocks out the row and column of a particular element. The cofactor is a slight twist on the minor: it's the minor of an element multiplied by \text{\((-1)^{i+j}\)}, with \text{\(i\)} and \text{\(j\)} being the row and column numbers of the element, respectively.

With minors and cofactors in hand, the cofactor expansion allows us to compute the determinant of a square matrix by taking any row or column, multiplying each element by its corresponding cofactor, and summing these products. The beauty of this method is that it applies to matrices of any size, enabling you to calculate the determinant in a manageable step-by-step manner, no matter how large the matrix.