Matrices

The One Thing

Matrices are rectangular arrays governed by order-dependent algebraic rules. For square matrices, the determinant determines invertibility: a non-zero determinant gives a unique inverse, which can be used to solve matrix equations and systems of simultaneous linear equations.

Definitions and Results

• Matrix: A rectangular array of elements arranged in rows and columns, usually written inside brackets.
• Order of a Matrix: A matrix with $m$ rows and $n$ columns has order $m\times n$.
• Element: An individual entry, denoted $a_{ij}$, where $i$ identifies the row and $j$ identifies the column.
• Row Matrix: A matrix having exactly one row.
• Column Matrix: A matrix having exactly one column.
• Square Matrix: A matrix with the same number of rows and columns.
• Zero or Null Matrix: A matrix in which every element is zero.
• Diagonal Matrix: A square matrix in which all non-diagonal elements are zero.
• Scalar Matrix: A diagonal matrix whose diagonal elements are all equal.
• Identity Matrix: A square matrix with $1$s on the principal diagonal and $0$s elsewhere. It is the multiplicative identity.
• Transpose: The transpose $A^{\mathrm{T}}$ is obtained by interchanging the rows and columns. If $A$ has order $m\times n$, then $A^{\mathrm{T}}$ has order $n\times m$.
• Symmetric Matrix: A square matrix $A$ satisfying $A^{\mathrm{T}}=A$.
• Skew-Symmetric Matrix: A square matrix $A$ satisfying $A^{\mathrm{T}}=-A$. Every diagonal element is therefore zero.
• Equality of Matrices: Two matrices are equal only when they have the same order and all corresponding elements are equal.
• Matrix Addition and Subtraction: Defined only for matrices of the same order and performed element by element.
• Scalar Multiplication: Every element of a matrix is multiplied by the scalar.
• Matrix Multiplication: $AB$ exists only when the number of columns of $A$ equals the number of rows of $B$. Each entry is obtained by multiplying a row of $A$ by a column of $B$ and adding the products.
• Determinant: A numerical value associated with a square matrix, used to test invertibility and solve matrix problems.
• Minor: The minor $M_{ij}$ is the determinant obtained by deleting the row and column containing $a_{ij}$.
• Cofactor:

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$

• Adjoint: The adjoint of a square matrix is the transpose of its cofactor matrix.
• Inverse Matrix: A square matrix $A$ has an inverse $A^{-1}$ if

$$A{A}^{-1}=A^{-1}A=I.$$

• Singular Matrix: A square matrix with determinant zero. It has no inverse.
• Non-Singular Matrix: A square matrix with non-zero determinant. It has a unique inverse.
• Matrix Equation: An equation involving matrices, such as $AX=B$, often solved using an inverse matrix.
• System of Linear Equations: A collection of linear equations represented as

$$AX=B,$$ where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix.

Important results include:

• Matrix addition is commutative:

$$A+B=B+A.$$

• Matrix multiplication is associative:

$$(AB)C=A(BC).$$

• Matrix multiplication is generally not commutative:

$$AB\ne BA.$$

• Matrix multiplication is distributive:

$$A(B+C)=AB+AC,(A+B)C=AC+BC.$$

• For a compatible matrix $A$,

$$AI=IA=A.$$

• For

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ $$\det (A)=ad-bc.$$

• A $3\times 3$ determinant may be expanded along any row or column using cofactors.
• Interchanging two rows or two columns changes the sign of a determinant.
• If two rows or two columns are identical or proportional, the determinant is zero.
• The determinant of a triangular matrix is the product of its diagonal elements.
• For square matrices of the same order,

$$\det (AB)=\det (A)\det (B),\det (A^{\mathrm{T}})=\det (A).$$

• A square matrix is invertible if and only if

$$\det (A)\ne 0.$$

• For a non-singular square matrix,

$$A^{-1}=\frac{\mathrm{adj}(A)}{\det (A)}.$$

• For

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ $$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],$$ provided $ad-bc\ne 0$.

• For matrices with existing inverses,

$$(AB{)}^{-1}=B^{-1}{A}^{-1}.$$

• The transpose of a product is

$$(AB{)}^{\mathrm{T}}=B^{\mathrm{T}}{A}^{\mathrm{T}}.$$

• If $AX=B$ and $A$ is invertible, then

$$X=A^{-1}B.$$

• If $XA=B$ and $A$ is invertible, then

$$X=B{A}^{-1}.$$

• Cramer’s rule applies to $n$ linear equations in $n$ unknowns when the determinant $\Delta$ of the coefficient matrix is non-zero:

$$x_i=\frac{{\Delta }_i}{\Delta }.$$

• A rank or consistency approach determines whether a system has a unique solution, infinitely many solutions, or no solution.
• Matrices are used in economics, computer graphics, coding, networks, statistics, transformations, and the solution of simultaneous equations.

Worked Methods

1. Adding and subtracting matrices

• Check that the matrices have the same order.
• Add or subtract corresponding elements.
• Write the resulting matrix in the same order.

Matrix addition and subtraction are not defined for matrices of different orders.

2. Multiplying by a scalar

• Identify the scalar.
• Multiply every element of the matrix by that scalar.
• Retain the original order of the matrix.

3. Multiplying two matrices

• Check that the number of columns of the first matrix equals the number of rows of the second.
• Select a row from the first matrix and a column from the second.
• Multiply corresponding entries.
• Add the products to obtain one entry of the product matrix.
• Repeat for every row-column combination.

The order of multiplication must be preserved. Reversing the factors generally changes the result or may make the product undefined.

4. Finding a transpose

• Interchange every row and column.
• Record the resulting matrix as $A^{\mathrm{T}}$.
• If $A$ is $m\times n$, verify that $A^{\mathrm{T}}$ is $n\times m$.
• Compare $A^{\mathrm{T}}$ with $A$ to test symmetry:

- $A^{\mathrm{T}}=A$: symmetric; - $A^{\mathrm{T}}=-A$: skew-symmetric.

5. Finding a $2\times 2$ determinant

For $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ calculate $$\det (A)=ad-bc.$$

The determinant is used to determine whether the matrix is singular or non-singular.

6. Finding a determinant using minors and cofactors

• Choose a row or column for expansion.
• For each element $a_{ij}$, delete its row and column.
• Find the resulting determinant $M_{ij}$, called the minor.
• Apply the sign factor:

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$

• Multiply each element by its cofactor.
• Add the resulting terms.

For a $3\times 3$ determinant, expansion may be carried out along any row or column.

7. Finding the inverse using the adjoint

• Confirm that the matrix is square.
• Calculate $\det (A)$.
• If $\det (A)=0$, the matrix is singular and has no inverse.
• Find each minor $M_{ij}$.
• Convert the minors into cofactors using

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$

• Form the cofactor matrix.
• Transpose the cofactor matrix to obtain $\mathrm{adj}(A)$.
• Use

$$A^{-1}=\frac{\mathrm{adj}(A)}{\det (A)}.$$

For $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ this gives $$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],$$ provided $ad-bc\ne 0$.

8. Solving a matrix equation

For $$AX=B:$$

• Confirm that $A$ is square and invertible.
• Multiply on the left by $A^{-1}$:

$$A^{-1}AX=A^{-1}B.$$

• Use $A^{-1}A=I$:

$$X=A^{-1}B.$$

For $$XA=B:$$

• Confirm that $A$ is invertible.
• Multiply on the right by $A^{-1}$:

$$XA{A}^{-1}=B{A}^{-1}.$$

• Use $A{A}^{-1}=I$:

$$X=B{A}^{-1}.$$

9. Solving simultaneous linear equations by matrices

• Express the system in the form

$$AX=B.$$

• Identify $A$ as the coefficient matrix, $X$ as the variable matrix, and $B$ as the constant matrix.
• Calculate $\det (A)$.
• If $\det (A)\ne 0$, find $A^{-1}$.
• Calculate

$$X=A^{-1}B.$$

Alternatively, use Cramer’s rule when there are $n$ equations in $n$ unknowns and the coefficient determinant is non-zero: $$x_i=\frac{{\Delta }_i}{\Delta }.$$

A rank or consistency approach may instead be used to establish whether the system has a unique solution, infinitely many solutions, or no solution.

Where It Goes Wrong

• Adding or subtracting matrices of different orders; these operations require corresponding matrices to have the same order.
• Multiplying matrices without checking compatibility; the number of columns of the first matrix must equal the number of rows of the second.
• Reversing matrix factors, despite the fact that matrix multiplication is generally not commutative:

$$AB\ne BA.$$

• Applying determinant or inverse formulas to rectangular matrices; determinants and inverses are defined here for square matrices.
• Forgetting that a zero determinant means the matrix is singular and has no inverse.
• Using $A^{-1}B$ for an equation of the form $XA=B$; the correct result is $X=B{A}^{-1}$.

What Gets Asked

• Define a matrix, its order, an element, and the principal types of matrices.
• Determine the order, transpose, or equality of given matrices.
• Add, subtract, or multiply matrices and justify whether an operation is defined.
• Test whether a matrix is symmetric or skew-symmetric.
• Calculate a $2\times 2$ determinant using

$$ad-bc.$$

• Expand a $3\times 3$ determinant along a specified row or column using minors and cofactors.
• Use determinant properties involving row interchange, identical or proportional rows, triangular matrices, products, and transposes.
• Determine whether a square matrix is singular or non-singular.
• Find an inverse using the adjoint formula or the $2\times 2$ inverse formula.
• Solve $AX=B$ or $XA=B$.
• Convert simultaneous linear equations into $AX=B$.
• Solve a system using the inverse-matrix method or Cramer’s rule.
• Use rank or consistency to classify a system as having a unique solution, infinitely many solutions, or no solution.
• Explain how matrices represent transformations or model relationships in economics, computer graphics, coding, networks, statistics, and simultaneous equations.