Use determinants to find the rank of the coefficient matrix of the following system of equations: seven 𝑥 plus two 𝑦 equals negative two and two 𝑥 plus five 𝑦 equals negative nine.
We will begin by writing the system of equations given in matrix form. If we have two equations in two unknowns as in this case, we start with a two-by-two coefficient matrix 𝑎, 𝑏, 𝑐, 𝑑. This is multiplied by a two-by-one variables matrix 𝑥, 𝑦, where 𝑥 and 𝑦 are the unknowns in the equations. The product of these is equal to the two-by-one constants matrix 𝑒, 𝑓, where 𝑒 and 𝑓 are the answer values in the two equations.
In this question, we have a coefficient matrix seven, two, two, five. Multiplying this by the variables matrix 𝑥, 𝑦 gives us the constants matrix negative two, negative nine. From this point, we are asked to find the rank of the coefficient matrix using determinants. We recall that 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. The process that we go through to find the rank of a two-by-two matrix 𝐴 is shown in the flow chart.
In this question, we will begin by letting 𝐴 be the coefficient matrix seven, two, two, five. By inspection, since our matrix is not the zero matrix, its rank is not equal to zero. Our next step is to calculate the determinant of matrix 𝐴. And we calculate the determinant of a two-by-two matrix by finding the product of the values in the top left and bottom right and subtracting the product of the values in the top right and bottom left. The determinant of matrix 𝐴 is therefore equal to seven multiplied by five minus two multiplied by two. This simplifies to 35 minus four, which is equal to 31. As the determinant is not equal to zero, the rank of the matrix is not equal to one. And we can therefore conclude that the rank of the coefficient matrix of the system of equations given is two.