# Question Video: Determining Whether the Number of Columns of a Coefficient Matrix in a Matrix Equation Represents the Number of Variables Mathematics • 10th Grade

True or False: The number of columns of a coefficient matrix in a matrix equation represents the number of variables.

02:17

### Video Transcript

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, letβs begin by considering some of the terms in the statement. First, we define a matrix equation. Thatβs an equation of the form π΄π is equal to π΅, where π΄ is the coefficient matrix, π is the variable matrix or matrix of variables, and π΅ is the constant matrix for a system of linear equations. Now, since the coefficient matrix is one of the terms in our statement, letβs consider what this represents. Suppose we have a system of linear equations where we have π equations in π unknown variables, where the πβs and πβs are constants and the π₯βs are the variable unknowns. Then the coefficient matrix π΄ is the matrix whose elements are the coefficients of each of the π variables in our π linear equations.

So, for example, the first row of the coefficient matrix has elements π one one, π one two, to π one π. And our second row has elements π two one, π two two, up to π two π, and so on to the πth row, which has elements π π one, π π two, all the way up to π ππ. The matrix π is the column matrix whose elements are the π variables. Thatβs π₯ one to π₯ π. And if π is multiplied by the coefficient matrix π΄, this gives us the left-hand side of our system of equations. Finally, the column matrix π΅ on the right-hand side is the constant matrix with elements π one to π π, 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 π΄, we see that this matrix has π columns. And π 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.