Matrices

The One Thing

Matrix calculations depend first on dimensions: addition and subtraction require equal orders, while multiplication requires the number of columns of the first matrix to equal the number of rows of the second. For a square matrix, invertibility is determined by the determinant: a matrix is invertible exactly when its determinant is non-zero.

Definitions and Results

• Matrix: A rectangular array of elements arranged in rows and columns, usually denoted by a capital letter such as $A$. If $A=[a_{ij}]$ is an $m\times n$ matrix, $i$ identifies the row and $j$ identifies the column of $a_{ij}$.

• Order of a Matrix: An $m\times n$ matrix has $m$ rows and $n$ columns.

• Row Matrix: A matrix with one row, such as $[abc]$.

• Column Matrix: A matrix with one column, such as $[a,b,c{]}^T$.

• Rectangular Matrix: A matrix with different numbers of rows and columns.

• Square Matrix: A matrix with the same number of rows and columns.

• Zero or Null Matrix: A matrix whose every element is zero. It is the additive identity:

$$A+O=A.$$ For every matrix $A$, $$A+(-A)=O.$$

• Diagonal Matrix: A square matrix whose elements outside the main diagonal are zero.

• Scalar Matrix: A diagonal matrix in which all main diagonal elements are equal.

• Identity Matrix: A square matrix with $1$s on the main diagonal and $0$s elsewhere, denoted by $I$. It satisfies

$$AI=IA=A$$ whenever the dimensions are compatible.

• Upper Triangular Matrix: A square matrix whose elements below the main diagonal are zero.

• Lower Triangular Matrix: A square matrix whose elements above the main diagonal are zero.

• Symmetric Matrix: A square matrix $A$ satisfying

$$A^T=A.$$

• Skew-Symmetric Matrix: A square matrix $A$ satisfying

$$A^T=-A.$$ Its diagonal elements must be zero, since $$a_{ii}=-a_{ii}.$$

• Equality of Matrices: Two matrices are equal only if they have the same order and all corresponding elements are equal.

• Matrix Addition and Subtraction: These operations are defined only for matrices of the same order and are performed element by element.

• Scalar Multiplication: Every element of a matrix is multiplied by the same scalar.

• Matrix Multiplication: $AB$ is defined 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 resulting products.

• Non-Commutativity of Multiplication: In general,

$$AB\ne BA.$$

• Associativity and Distributivity: Whenever the products are defined,

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

• Transpose: $A^T$ is obtained by interchanging the rows and columns of $A$. If $A$ is $m\times n$, then $A^T$ is $n\times m$. The main transpose identities are

$$(A^T{)}^T=A,$$ $$(A+B{)}^T=A^T+B^T,$$ $$(kA{)}^T=k{A}^T,$$ $$(AB{)}^T=B^T{A}^T.$$

• Symmetric and Skew-Symmetric Decomposition: Every square matrix can be expressed uniquely as

$$A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T),$$ where the first term is symmetric and the second is skew-symmetric.

• Determinant: A numerical value associated with a square matrix that helps determine whether it is invertible. For

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

• Invertible or Non-Singular Matrix: A square matrix is invertible if

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

• Singular Matrix: A square matrix is singular if

$$\det (A)=0.$$ A singular matrix has no inverse.

• Inverse Matrix: $A^{-1}$ is the inverse of a square matrix $A$ if

$$A{A}^{-1}=A^{-1}A=I.$$ The inverse is unique.

• Adjoint: The adjoint of $A$, written $\mathrm{adj}(A)$, is the transpose of its cofactor matrix and is used to calculate the inverse.

• Inverse Formula: If $\det (A)\ne 0$, then

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

• Cofactor and Adjoint for a $2\times 2$ Matrix: For

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ the cofactor matrix is $$\left[\begin{array}{cc}d& -c\\ -b& a\end{array}\right]$$ after arranging cofactors appropriately, and $$\mathrm{adj}(A)=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right].$$

• Inverse Identities: If $A$ and $B$ are invertible, then

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

• Triangular Determinant: The determinant of a triangular matrix is the product of its diagonal elements.

• Elementary Row or Column Operations: These include interchanging rows or columns, multiplying a row or column by a non-zero scalar, and adding a multiple of one row or column to another.

• Matrix Equation: To solve

$$AX=B,$$ when $A$ is invertible, multiply on the left by $A^{-1}$: $$X=A^{-1}B.$$ Systems of linear equations may be represented in matrix form and solved using inverse matrices, determinants, or elementary operations.

Worked Methods

Classifying a Matrix

• Count the rows and columns.
• State the order $m\times n$.
• Identify whether the matrix is a row, column, rectangular, or square matrix.
• Inspect its entries to determine whether it is zero, diagonal, scalar, identity, upper triangular, lower triangular, symmetric, or skew-symmetric.
• For symmetry, calculate or compare the transpose:

$$A^T=A.$$

• For skew-symmetry, verify

$$A^T=-A$$ and confirm that every diagonal element is zero.

Adding or Subtracting Matrices

• Check that the matrices have the same order.
• Add or subtract corresponding elements.
• If the orders differ, the operation is not defined.

The zero matrix acts as the additive identity: $$A+O=A.$$

Multiplying by a Scalar

• Take the given scalar $k$.
• Multiply every element of the matrix by $k$.
• The order of the matrix remains unchanged.

The transpose obeys $$(kA{)}^T=k{A}^T.$$

Multiplying Two Matrices

• Check that the number of columns of $A$ equals the number of rows of $B$.
• Form each entry of $AB$ by taking a row of $A$ and the corresponding column of $B$.
• Multiply corresponding entries.
• Add the products.
• Repeat for every row-column pair.
• Do not assume that $AB=BA$; matrix multiplication is generally not commutative.

Matrix multiplication remains associative and distributive: $$(AB)C=A(BC),$$ $$A(B+C)=AB+AC,$$ $$(A+B)C=AC+BC.$$

Finding a Transpose

• Write the rows of $A$ as the columns of $A^T$.
• If $A$ is $m\times n$, record that $A^T$ is $n\times m$.
• Apply the relevant identities where required:

$$(A+B{)}^T=A^T+B^T,$$ $$(AB{)}^T=B^T{A}^T.$$

• The order of multiplication reverses when transposing a product.

Decomposing a Square Matrix

For a square matrix $A$:

• Form $A+A^T$.
• Multiply by $\frac{1}{2}$:

$$\frac{1}{2}(A+A^T).$$ This is symmetric.

• Form $A-A^T$.
• Multiply by $\frac{1}{2}$:

$$\frac{1}{2}(A-A^T).$$ This is skew-symmetric.

• Add the two parts:

$$A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T).$$

Finding the Determinant of a $2\times 2$ Matrix

For $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$

• Multiply the main-diagonal entries:

$$ad.$$

• Multiply the other diagonal entries:

$$bc.$$

• Subtract:

$$\det (A)=ad-bc.$$

• If $\det (A)\ne 0$, the matrix is invertible.
• If $\det (A)=0$, the matrix is singular and has no inverse.

Finding the Inverse of a $2\times 2$ Matrix

For $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$

• Calculate

$$\det (A)=ad-bc.$$

• If the determinant is zero, stop: $A$ has no inverse.
• Form the adjoint:

$$\mathrm{adj}(A)=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right].$$

• Multiply the adjoint by the reciprocal of the determinant:

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

• The result satisfies

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

Solving a Matrix Equation

To solve $$AX=B$$ when $A$ is invertible:

• Establish that $A$ is square and $\det (A)\ne 0$.
• Multiply both sides on the left by $A^{-1}$:

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

• Use $A^{-1}A=I$:

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

• A system of linear equations can therefore be represented in matrix form and solved by inverse matrices, determinants, or elementary operations.

Where It Goes Wrong

• Adding or subtracting matrices of different orders is invalid; the matrices must have the same order.
• Multiplication is often attempted without checking dimensions; the number of columns of the first matrix must equal the number of rows of the second.
• The order of matrix multiplication is frequently reversed incorrectly: generally $AB\ne BA$, and

$$(AB{)}^T=B^T{A}^T,$$ not $A^T{B}^T$.

• An inverse is calculated for a non-square matrix or for a square matrix with determinant zero; only square matrices with non-zero determinant are invertible.
• A skew-symmetric matrix is identified without checking its diagonal entries; every diagonal element must be zero.
• In the $2\times 2$ inverse formula, the signs and positions in

$$\mathrm{adj}(A)=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$ must be preserved, and division by $\det (A)$ is valid only when the determinant is non-zero.

What Gets Asked

This material supports questions requiring students to:

• State the order of a matrix and classify it as a row, column, rectangular, square, zero, diagonal, scalar, identity, upper triangular, lower triangular, symmetric, or skew-symmetric matrix.
• Determine whether two matrices are equal.
• Add, subtract, or multiply matrices after checking their dimensions.
• Apply scalar multiplication and the identity and zero matrix properties.
• Calculate a matrix product using row-column products.
• Explain why matrix multiplication is associative and distributive but generally not commutative.
• Find the transpose of a matrix and use

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

• Decompose a square matrix into symmetric and skew-symmetric parts.
• Calculate the determinant of a $2\times 2$ matrix using

$$ad-bc.$$

• Find the cofactor matrix and adjoint of

$$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right].$$

• Determine whether a matrix is singular or non-singular.
• Find the inverse using

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

• Use triangular diagonal entries to calculate a determinant.
• Verify inverse and transpose identities.
• Solve $AX=B$ using

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

• Represent and solve systems of linear equations using inverse matrices, determinants, or elementary row and column operations.