Question Video: Determining the Number of Equations in a Linear System from the Order of the Coefficient Matrix Mathematics • Third Year of Secondary School

Video Transcript

Consider a system of linear equations written in matrix form as 𝐴𝑋 is equal to 𝐵. If the order of matrix 𝐴 is 𝑚 by 𝑛 and the order of matrix 𝑋 is 𝑛 by one, how many equations does the system have?

We’re asked how many equations does a system of linear equations have using the fact that, in the matrix equation 𝐴𝑋 is equal to 𝐵, the matrix 𝐴 has order 𝑚 by 𝑛 and the matrix 𝑋 has order 𝑛 by one. The matrix 𝐴 is called the coefficient matrix. The matrix 𝑋 is the variable matrix. And on the right-hand side, the matrix 𝐵 is the constant matrix.

We can write this matrix equation as shown. And if we multiply our 𝑚-by-𝑛 matrix 𝐴 by the column matrix 𝑋, we recover our system of linear equations. And we see that there are actually 𝑚 equations. The order of the matrix 𝐴 then is the number of equations 𝑚 multiplied by the number of unknowns 𝑛. Our answer then is that the system has 𝑚 equations.