Key takeaway: This precalculus matrix systems lesson turns a set of linear equations into one compact statement, \(AX=B\). The matrix \(A\) holds the coefficients, \(X\) holds the unknowns, and \(B\) holds the constants. When the determinant is not zero, the inverse gives the unique answer \(X=A^{-1}B\). Standards A-REI.C.8 and N-VM.C.12 cover the work. The final sections show how the same matrices move points and scale area.
A matrix equation \(AX=B\) packs a linear system into one statement, where \(A\) holds the coefficients, \(X\) the unknowns and \(B\) the constants. When \(\det A\neq 0\), the inverse gives the unique solution \(X=A^{-1}B\). The same matrices also transform the plane, and \(|\det A|\) is the factor by which areas change.
Precalculus Matrix Systems Lesson: key facts
- A system of linear equations can be written as \(AX=B\).
- For a \(2\times 2\) matrix, \(\det A=ad-bc\) and \(A^{-1}=\frac{1}{ad-bc}\) times the matrix with rows \(d,\ -b\) and \(-c,\ a\).
- If \(\det A\neq 0\), then \(X=A^{-1}B\) is the only solution; if \(\det A=0\), there is no inverse.
- The columns of a \(2\times 2\) matrix are the images of \((1,\ 0)\) and \((0,\ 1)\).
- A transformation by \(M\) multiplies every area by \(|\det M|\).
How do you write a system as a matrix equation?
A system of linear equations packs into one equation, \(AX=B\). The coefficient matrix \(A\) lists the coefficients, the column \(X\) holds the unknowns, and the column \(B\) holds the constants. The system \(2x+3y=8\), \(x-y=-1\) becomes the product below.
How do you find the inverse of a 2 by 2 matrix?
For a matrix with first row \(a,\ b\) and second row \(c,\ d\), the determinant is \(ad-bc\). When it is not zero, the inverse has first row \(d,\ -b\) and second row \(-c,\ a\), all multiplied by \(\frac{1}{ad-bc}\). Then \(AA^{-1}=A^{-1}A=I\), where \(I\) is the identity matrix with rows \(1,\ 0\) and \(0,\ 1\).
If \(ad-bc=0\), the matrix is singular and has no inverse.
How do you solve AX = B with the inverse?
- Write the system as \(AX=B\).
- Compute \(\det A\). If it is \(0\), stop: there is no unique solution.
- Build \(A^{-1}\).
- Multiply on the left: \(X=A^{-1}B\). Order matters, because matrix multiplication is not commutative.
- Check the answer in both original equations.
Example. For \(A\) above, \(\det A=2(-1)-3(1)=-5\). So \(A^{-1}\) has rows \(\frac{1}{5},\ \frac{3}{5}\) and \(\frac{1}{5},\ -\frac{2}{5}\). Then \(x=\frac{1}{5}(8)+\frac{3}{5}(-1)=1\) and \(y=\frac{1}{5}(8)-\frac{2}{5}(-1)=2\). Check: \(2(1)+3(2)=8\) and \(1-2=-1\). The solution is \((1,\ 2)\).
For a \(3\times 3\) or larger system, a graphing calculator or software finds \(A^{-1}\); the method \(X=A^{-1}B\) stays the same.
How does a matrix transform the plane?
Multiplying a point \((x,\ y)\), written as a column, by a \(2\times 2\) matrix \(M\) moves it to a new point. The two columns of \(M\) are the images of \((1,\ 0)\) and \((0,\ 1)\). For \(M\) with columns \((2,\ 0)\) and \((1,\ 3)\), the unit square lands on the parallelogram below. For example, \((1,\ 1)\) goes to \((2+1,\ 0+3)=(3,\ 3)\).
What does the determinant say about area?
The image of any region has area \(|\det M|\) times the original area. Here \(\det M=2\cdot 3-1\cdot 0=6\), so the unit square (area \(1\)) becomes a parallelogram of area \(6\). A negative determinant flips orientation without changing the area rule. A determinant of \(0\) squashes the plane onto a line, so no inverse can undo it.
Solving Systems with Matrices: frequently asked questions
What is the inverse of a matrix?
The inverse \(A^{-1}\) is the matrix that undoes \(A\), so that \(AA^{-1}=A^{-1}A=I\). Only square matrices with a nonzero determinant have one.
Why does the order in X = A^{-1}B matter?
Matrix multiplication is not commutative. To cancel \(A\) from \(AX=B\), multiply on the left of both sides by \(A^{-1}\), which gives \(X=A^{-1}B\), not \(BA^{-1}\).
What happens when the determinant is zero?
The matrix has no inverse. The lines of a 2 by 2 system are then parallel (no solution) or the same line (infinitely many solutions).
How do you solve a 3 by 3 system with matrices?
Write it as \(AX=B\), enter \(A\) and \(B\) into a graphing calculator or software, compute \(A^{-1}\), and multiply \(A^{-1}B\). By hand the method is long, so technology is used.
What does the determinant mean geometrically?
For a \(2\times 2\) matrix, \(|\det|\) is the area of the image of the unit square, and so the area scale factor of the whole transformation. A negative sign means the plane is flipped.
How do you write a system as a matrix equation?
Line up the variables and read off the coefficients. For \(3x+y=7\) and \(2x-y=3\), the coefficient matrix has rows \(3,1\) and \(2,-1\). The unknown column is \(\begin{pmatrix}x\\y\end{pmatrix}\), and the constants are \(7\) and \(3\). Keeping the variable order identical in every equation prevents sign slips.
How do you solve AX = B with an inverse?
First, find \(\det A=ad-bc\). If it is nonzero, the inverse is \(\frac{1}{ad-bc}\) times the matrix with rows \(d,-b\) and \(-c,a\). Then multiply on the left: \(X=A^{-1}B\). For \(3x+4y=10\) and \(2x+3y=7\), the determinant is 1, and the solution is \(x=2\), \(y=1\). Order matters here, because \(BA^{-1}\) is a different product.
What happens when the determinant is zero?
No inverse exists, so the method stops. The system \(x+2y=4\) and \(3x+6y=12\) has \(\det A=0\), yet both equations describe the same line. As a result, it has infinitely many solutions. A parallel pair would have none.
How does a matrix move the plane?
Multiplying a point by a matrix moves it to a new place. The matrix with rows \(1,2\) and \(0,3\) sends \((2,-3)\) to \((-4,-9)\). Its determinant, 3, is the area scale factor, and a negative determinant also flips orientation. In other words, a region of area 5 under a determinant of \(-3\) becomes area 15.
What practice comes next?
Pick two equations from a real purchase, such as two shopping trips with different item counts. Write them as \(AX=B\), compute the determinant, and solve. Then multiply back to check that both original equations hold.
See every chapter on the Precalculus math page.
