Video Transcript
Express the simultaneous equations
three ๐ plus two ๐ equals 13 and two ๐ plus three ๐ equal seven as a matrix
equation.
We recall that we can represent a
system of linear equations in matrix form using a coefficient matrix, a variable
matrix, and a constant matrix. Before doing this, we need to
ensure that our equations are written in standard form, ๐๐ฅ plus ๐๐ฆ is equal to
๐, where ๐, ๐, and ๐ are constants. In this question, both of our
equations are written in standard form. However, it is important to note
that ๐ and ๐ are variables and not constants.
We know that the coefficient matrix
can be formed by aligning the coefficients of the variables of each equation in a
row. The coefficients of our first
equation are three and two. In the second equation, we have two
and three, giving us the two-by-two coefficient matrix three, two, two, three. The variables here are ๐ and
๐. So we can write the variable matrix
as the two-by-one matrix ๐, ๐. On the right-hand side, we have the
constant terms 13 and seven. As these correspond to the first
and second equation, respectively, the constant matrix is 13, seven.
We now have a matrix equation made
up of a coefficient matrix, a variable matrix, and a constant matrix. The simultaneous equations three ๐
plus two ๐ is equal to 13 and two ๐ plus three ๐ is equal to seven can be
rewritten as a matrix equation three, two, two, three multiplied by ๐, ๐ is equal
to 13, seven.