We want to represent this system by using the matrix equationAX=B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. To do so, we need to write the variable terms in the same order. Let's do it by writing zero for the missing variables' coefficients.
⎩⎪⎪⎨⎪⎪⎧1x+0y+2z=-10x+1y−2z=22x+1y+1z=1
Great! Now we can write the system as a matrix equation.
⎣⎢⎡1x+0y+2z0x+1y−2z2x+1y+1z⎦⎥⎤=⎣⎢⎡-121⎦⎥⎤
Next, we will write the left-hand side of the above matrix equation as the product of the coefficient matrix and the variable matrix, while the right-hand side is the constant matrix.
⎣⎢⎡1020112-21⎦⎥⎤⋅⎣⎢⎡xyz⎦⎥⎤=⎣⎢⎡-121⎦⎥⎤
We finally showed the given system in a matrix equation form.
b We will now find the determinant of matrix A, which we found in Part A. To do so we need to apply the following steps. First we will copy the first and second columns and write them on the right-hand side.
Second, we need to draw diagonals to show the terms that will be multiplied. Three diagonals will begin with the upper-left numbers and move down to the right. Another three will begin with the bottom-left numbers and move up to the right.
c Last, we will solve the matrix equation that we wrote in Part A.
⎣⎢⎡1020112-21⎦⎥⎤⋅⎣⎢⎡xyz⎦⎥⎤=⎣⎢⎡-121⎦⎥⎤
To do so we first need to have the inverse of A. Since we know that detA=0 from Part B, we can find A-1. Let's use a calculator. To create a matrix, we push 2ND and x-1. Then, using the right arrow, we navigate to the Edit tab and push ENTER. The matrix is entered by defining the dimensions, 3×3, and inputting the elements.
Now that we have entered the matrix, we push 2ND followed by MODE to go back to the main screen. Then, we push 2ND and x-1, select our matrix, and push ENTER.
Finally, we push x-1 followed by ENTER. This will give us the inverse of our matrix!
Great! Then we will multiply both sides of the matrix equation by A-1 to isolate the variable matrix.
A-1AX=A-1B⇒X=A-1B
Since the product of A-1A is equal to the identity matrix, it is enough to just multiply A-1 by B to find the variable matrix X. Let's do it!
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