True or False: The number of columns of a coefficient matrix in a matrix equation represents the number of variables.
To determine whether the given statement is true or false, let’s begin by considering some of the terms in the statement. First, we define a matrix equation. That’s an equation of the form 𝐴𝑋 is equal to 𝐵, where 𝐴 is the coefficient matrix, 𝑋 is the variable matrix or matrix of variables, and 𝐵 is the constant matrix for a system of linear equations. Now, since the coefficient matrix is one of the terms in our statement, let’s consider what this represents. Suppose we have a system of linear equations where we have 𝑚 equations in 𝑛 unknown variables, where the 𝑎’s and 𝑏’s are constants and the 𝑥’s are the variable unknowns. Then the coefficient matrix 𝐴 is the matrix whose elements are the coefficients of each of the 𝑛 variables in our 𝑚 linear equations.
So, for example, the first row of the coefficient matrix has elements 𝑎 one one, 𝑎 one two, to 𝑎 one 𝑛. And our second row has elements 𝑎 two one, 𝑎 two two, up to 𝑎 two 𝑛, and so on to the 𝑚th row, which has elements 𝑎 𝑚 one, 𝑎 𝑚 two, all the way up to 𝑎 𝑚𝑛. The matrix 𝑋 is the column matrix whose elements are the 𝑛 variables. That’s 𝑥 one to 𝑥 𝑛. And if 𝑋 is multiplied by the coefficient matrix 𝐴, this gives us the left-hand side of our system of equations. Finally, the column matrix 𝐵 on the right-hand side is the constant matrix with elements 𝑏 one to 𝑏 𝑚, representing the right-hand side of our system of linear equations.
Now, looking back at our statement, this says the number of columns of a coefficient matrix in a matrix equation represents the number of variables. And if we look at our coefficient matrix 𝐴, we see that this matrix has 𝑛 columns. And 𝑛 is indeed the number of variables in our system of equations. The given statement is therefore true. The number of columns of a coefficient matrix in a matrix equation does represent the number of variables.