Use determinants to find the rank of the augmented matrix of the following system of equations. Two 𝑥 plus four 𝑦 equals negative three and two 𝑥 plus three 𝑦 equals negative six.
We will begin by identifying the augmented matrix from the system of equations. An augmented matrix has two parts. Firstly, since we have two equations in two unknowns, we begin with the two-by-two coefficient matrix 𝑎, 𝑏, 𝑐, 𝑑. The second part of our augmented matrix contains the constants on the right-hand side of our equations.
In this question, the coefficient matrix is equal to two, four, two, three. And the constants on the right-hand side of our equations are negative three and negative six. We will let 𝐴 be the two-by-three matrix two, four, negative three, two, three, negative six. We are asked to find the rank of this matrix using determinants. And we recall the rank of a matrix 𝐴 written RK of 𝐴 is the number of rows or columns 𝑛 of the largest 𝑛-by-𝑛 square submatrix of 𝐴 for which the determinant is nonzero. As the augmented matrix is a two-by-three matrix, we will need to consider two-by-two square submatrices of 𝐴.
We can find these submatrices by deleting one of the columns from matrix 𝐴. And we will then calculate the determinant of the remaining two-by-two matrix. It is important to note that if this determinant is zero, we need to repeat the process for the other two-by two sub matrices of 𝐴. We would do this by deleting the first column and then the second column of matrix 𝐴. When dealing with two-by-two matrices, the flowchart shown is a useful visual aid to help us determine the rank. We can see by inspection that the matrix two, four, two, three is not the zero matrix. Therefore, the rank is not equal to zero.
To calculate the determinant of this matrix, we find the product of the elements in the top left and bottom right and subtract the product of the elements in the top right and bottom left. In this question, this is equal to two multiplied by three minus four multiplied by two. This simplifies to six minus eight, which is equal to negative two. The determinant of our two-by-two matrix is not equal to zero.
We can therefore conclude that the rank of the augmented matrix of the system of equations given is two.