Video Transcript
Use matrices to solve the system
negative 𝑥 plus five 𝑦 equals eight, negative three 𝑥 plus 𝑦 equals eight.
We can represent this system of two
equations in two unknowns as a matrix equation. We begin by recalling that the two
equations 𝑎𝑥 plus 𝑏𝑦 equals 𝑒 and 𝑐𝑥 plus 𝑑𝑦 equals 𝑓 can be written as a
matrix equation as shown. We begin with the two-by-two
coefficient matrix 𝑎, 𝑏, 𝑐, 𝑑. This is multiplied by the variable
matrix 𝑥, 𝑦 and is equal to the constants matrix 𝑒, 𝑓.
In this question, the coefficient
matrix is negative one, five, negative three, one. Multiplying this by the variable
matrix 𝑥, 𝑦 gives us the constants matrix eight, eight. We can solve this matrix equation
by multiplying from the left the inverse of the coefficient matrix if it exists. We know that the inverse of a
square matrix exists if its determinant is not equal to zero. Let us first compute the
determinant of this matrix.
We know that the determinant of the
matrix 𝑎, 𝑏, 𝑐, 𝑑 is equal to 𝑎𝑑 minus 𝑏𝑐. Applying this formula to the
coefficient matrix, we have negative one multiplied by one minus five multiplied by
negative three, which is equal to 14. Since the determinant is nonzero,
we can proceed to find its inverse. We recall the formula for the
inverse of a two-by-two matrix 𝐴 equal to 𝑎, 𝑏, 𝑐, 𝑑 is one over the
determinant of 𝐴 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎. Hence, using the determinant of the
coefficient matrix, we see that its inverse is one over 14 multiplied by one,
negative five, three, negative one.
Next, we recall that for any
invertible matrix 𝐴, we have 𝐴 inverse multiplied by 𝐴 is equal to the identity
matrix 𝐼. This means that we’ll be able to
remove the coefficient matrix from the left-hand side of our equation by multiplying
from the left the inverse of the coefficient matrix. Multiplying both sides of the
equation by the inverse of the coefficient matrix, we have 𝑥, 𝑦 is equal to one
over 14 multiplied by one, negative five, three, negative one multiplied by eight,
eight.
We know that in order to multiply a
pair of matrices, the number of columns of the first matrix must equal the number of
rows of the second matrix. We can see that the matrix
multiplication on the right-hand side of the equation is well defined. Computing this matrix
multiplication, we obtain one multiplied by eight plus negative five multiplied by
eight and three multiplied by eight plus negative one multiplied by eight. So 𝑥, 𝑦 is equal to one
fourteenth of this. The right-hand side of our equation
simplifies to one over 14 multiplied by negative 32, 16. Finally, computing the scalar
multiplication, we have 𝑥, 𝑦 is equal to negative 16 over seven, eight over seven,
which is the solution of the matrix equation.
We know that a pair of matrices are
equal if each pair of corresponding entries are equal. Hence, we obtain the solution to
the given system of equations: 𝑥 is equal to negative 16 over seven and 𝑦 is equal
to eight over seven.
