System of Equations With Matrices: Cheat Sheet

Essential Concepts

Introduction to Matrices and Matrix Operations

  • A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
  • The dimensions of a matrix refer to the number of rows and the number of columns. A [latex]3\times 2[/latex] matrix has three rows and two columns.
  • Each number in the matrix is called an element or entry in the matrix.
  • A square matrix is a matrix that has the same number of rows as columns.
  • A row matrix is a matrix consisting of one row with dimensions [latex]1\text{ }\times \text{ }n[/latex]: [latex]\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right][/latex]
  • A column matrix is a matrix consisting of one column with dimensions [latex]m\text{ }\times \text{ }1[/latex]: [latex]\left[\begin{array}{c}{a}_{11}\\ {a}_{21}\\ {a}_{31}\end{array}\right][/latex]
  • We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix.
  • Scalar multiplication involves multiplying each entry in a matrix by a constant.
  • Scalar multiplication is often required before addition or subtraction can occur.
  • Scalar multiplication involves multiplying each entry in a matrix by a constant. A scalar multiple is any entry of a matrix that results from scalar multiplication. Scalar multiplication is distributive.
  • Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
  • Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix.
  • In matrix multiplication, we calculate each element of the resulting matrix by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. This involves multiplying corresponding entries and then summing those products.That is:For [latex]A=\left[\begin{array}{rrr}\hfill {a}_{11}& \hfill {a}_{12}& \hfill {a}_{13}\\ \hfill {a}_{21}& \hfill {a}_{22}& \hfill {a}_{23}\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}\hfill {b}_{11}& \hfill {b}_{12}& \hfill {b}_{13}\\ \hfill {b}_{21}& \hfill {b}_{22}& \hfill {b}_{23}\\ \hfill {b}_{31}& \hfill {b}_{32}& \hfill {b}_{33}\end{array}\right][/latex],

    [latex]AB=\left[\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}\\ \end{array}\\ {a}_{21}\cdot {b}_{11}+{a}_{22}\cdot {b}_{21}+{a}_{23}\cdot {b}_{31}\end{array}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}\\ \end{array}\\ {a}_{21}\cdot {b}_{12}+{a}_{22}\cdot {b}_{22}+{a}_{23}\cdot {b}_{32}\end{array}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}\\ \end{array}\\ {a}_{21}\cdot {b}_{13}+{a}_{22}\cdot {b}_{23}+{a}_{23}\cdot {b}_{33}\end{array}\right][/latex]

Solving System of Equations using Matrices

  • An augmented matrix is one that contains the coefficients and constants of a system of equations.
  • A matrix augmented with the constant column can be represented as the original system of equations.
  • Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
  • Row echelon form is a specific arrangement of a matrix used to solve systems of linear equations. In this form, each row has a leading entry of [latex]1[/latex] (called a leading [latex]1[/latex]) that appears to the right of the leading [latex]1[/latex] in the row above it. All entries below each leading [latex]1[/latex] are zeros.
  • The Gaussian elimination method refers to a strategy used to obtain the row-echelon form of a matrix.
  • We can use Gaussian elimination to solve a system of equations.
  • To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions.

Solving Systems with Inverses

  • An identity matrix has the property [latex]AI=IA=A[/latex].
  • If [latex]A[/latex] is an [latex]n\times n[/latex] matrix and [latex]B[/latex] is an [latex]n\times n[/latex] matrix such that [latex]AB=BA={I}_{n}[/latex], then [latex]B={A}^{-1}[/latex], the multiplicative inverse of a matrix [latex]A[/latex].
  • An invertible matrix has the property [latex]A{A}^{-1}={A}^{-1}A=I[/latex].
  • Use matrix multiplication and the identity to find the inverse of a [latex]2\times 2[/latex] matrix.
  • An identity matrix has the property [latex]AI=IA=A[/latex].
  • Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication.
  • Another way to find the multiplicative inverse is by augmenting with the identity. When matrix [latex]A[/latex] is transformed into [latex]I[/latex], the augmented matrix [latex]I[/latex] transforms into [latex]{A}^{-1}[/latex].
  • We can augment a [latex]3\times 3[/latex] matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse.
  • If [latex]A[/latex] is a [latex]2\times 2[/latex] matrix, such as

    [latex]A=\left[\begin{array}{rrr}\hfill a& \hfill & \hfill b\\ \hfill c& \hfill & \hfill d\end{array}\right][/latex]

    the multiplicative inverse of [latex]A[/latex] is given by the formula

    [latex]{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{rrr}\hfill d& \hfill & \hfill -b\\ \hfill -c& \hfill & \hfill a\end{array}\right][/latex]

    where [latex]ad-bc\ne 0[/latex]. If [latex]ad-bc=0[/latex], then [latex]A[/latex] has no inverse.

  • To solve a system of linear equations using an inverse matrix, let [latex]A[/latex] be the coefficient matrix, let [latex]X[/latex] be the variable matrix, and let [latex]B[/latex] be the constant matrix. Thus, we want to solve a system [latex]AX=B[/latex].

Key Equations

Identity matrix for a [latex]2\text{}\times \text{}2[/latex] matrix [latex]{I}_{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right][/latex]
Identity matrix for a [latex]\text{3}\text{}\times \text{}3[/latex] matrix [latex]{I}_{3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right][/latex]
Multiplicative inverse of a [latex]2\text{}\times \text{}2[/latex] matrix [latex]{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],\text{ where }ad-bc\ne 0[/latex]

Glossary

augmented matrix
a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets
coefficient matrix
a matrix that contains only the coefficients from a system of equations
column
a set of numbers aligned vertically in a matrix
entry
an element, coefficient, or constant in a matrix
Gaussian elimination
using elementary row operations to obtain a matrix in row-echelon form
identity matrix
a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra
main diagonal
entries from the upper left corner diagonally to the lower right corner of a square matrix

matrix
a rectangular array of numbers

multiplicative inverse of a matrix
a matrix that, when multiplied by the original, equals the identity matrix
row
a set of numbers aligned horizontally in a matrix
row-echelon form
after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal
row-equivalent
two matrices [latex]A[/latex] and [latex]B[/latex] are row-equivalent if one can be obtained from the other by performing basic row operations
row operations
adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form
scalar multiple
an entry of a matrix that has been multiplied by a scalar