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.
