Matrices and Determinants

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Introduction

MATH1012 Chapter 3 Goals

• Understand the operations of matrix algebra and identify the similarities and differences between matrix algebra and the algebra of real numbers
• Describe, and find bases for, the row space, column space and null space of a matrix
• Find the rank and nullity of a matrix and understand how they are related by the rank-nullity theorem
• Compute determinants and understand the relationship between determinants, rank and invertibility of matrices

• Intro Video - 3Blue1Brown:
• Textbook → Cambridge Matrices
• Matrices are a rectangular arrangement of numbers into rows and columns
  • Matrices are used to represent linear transformations
    • Vectors can be thought as matrices with one column
  • Numbers can be thought as vectors with one component and thus vectors can be thought as matrices with one column
    

    Transclude of matricescircle

Matrix Rules

• A matrix is a rectangular array of mathematical objects (usually numbers)
  • E.g. $A=(\genfrac{}{}{0ex}{}{3}{5}\genfrac{}{}{0ex}{}{-1}{12}\genfrac{}{}{0ex}{}{0}{-4})$
    • Matrices are represented with capital letters, such as $A$
  • If it has $m$ rows and $n$ columns, it is called a $m\times n$ matrix
    • The matrix above is a $2\times 3$ matrix
  • Entries in can be indexed using subscripts for the row and column
    • E.g. $a_{13}=0$ in the matrix above
      • $1$ is the row, $3$ is the column
      • A lower case letter is used for a term
  • Matrices are only equal if they have the same number of rows and columns and all corresponding entries are equal

Matrix Algebra

Matrix Addition and Subtraction

• Two matrices can be added or subtracted by adding or subtracting the corresponding entries
  • Example:
    • If $A=(\genfrac{}{}{0ex}{}{1}{-2}\genfrac{}{}{0ex}{}{6}{5})$ and $B=(\genfrac{}{}{0ex}{}{2}{3}\genfrac{}{}{0ex}{}{-8}{0})$
    • Then $A+B=(\genfrac{}{}{0ex}{}{1+2}{-2+3}\genfrac{}{}{0ex}{}{6+(-8)}{5+0})=(\genfrac{}{}{0ex}{}{3}{1}\genfrac{}{}{0ex}{}{-2}{5})$
  • Matrices can only be added/subtracted if they are of the same size
    • $m_1=m_2,n_1=n_2$

The Zero Matrix

• The $m\times n$ with all entries equal to zero is called the zero matrix and will be denoted with $O$
• For any $m\times n$ matrix $A$ and the $m\times n$ zero matrix $O$, we have:
  • $A+O=A$ and $A+(-A)=O$

Scalar Multiplication

• To multiply by a scalar, multiply each entry by the scalar
  • Scalars are real numbers
• Example:
  • If $A=(\genfrac{}{}{0ex}{}{1}{-2}\genfrac{}{}{0ex}{}{6}{5})$, then $3A=(\genfrac{}{}{0ex}{}{3\times 1}{3\times -2}\genfrac{}{}{0ex}{}{3\times 6}{3\times 5})=(\genfrac{}{}{0ex}{}{3}{-6}\genfrac{}{}{0ex}{}{18}{15})$

Matrix Multiplication

• Matrix multiplication is more complex than the other topics we have seem so far
  • Example of matrix multiplication:
    • Let $A=[\genfrac{}{}{0ex}{}{1}{4}\genfrac{}{}{0ex}{}{3}{2}]$ and $B=[\genfrac{}{}{0ex}{}{5}{6}\genfrac{}{}{0ex}{}{1}{0}]$
    • $A\times B=AB=[\genfrac{}{}{0ex}{}{1}{4}\genfrac{}{}{0ex}{}{3}{2}][\genfrac{}{}{0ex}{}{5}{6}\genfrac{}{}{0ex}{}{1}{0}]$
    • $[\genfrac{}{}{0ex}{}{1\times 5+3\times 6}{4\times 5+2\times 6}\genfrac{}{}{0ex}{}{1\times 1+3\times 0}{4\times 1+2\times 0}]$
    • $[\genfrac{}{}{0ex}{}{23}{32}\genfrac{}{}{0ex}{}{1}{4}]$
  • Matrix multiplication takes the values in $A$ going across in each row and multiplying them by the values in $B$ going down the corresponding column
    • $a_{11}\times b_{11},a_{12}\times b_{21}$
    • These values are added together to get the value in matrix $C$
• $A$ has dimensions $m\times n$ and $B$ has dimensions $p\times q$, if $n=p$ they can be multiplied
  • The resultant matrix is $m\times q$ in size
    • e.g. a matrix of size $2\times 3$ multiplied by a matrix $3\times 4$ will have a size of $2\times 4$, the like terms are removed similar to vector addition
  • Matrices can only be multiplied if the number or rows for $A$ = the number of columns for $B$
• The order is very important for matrix multiplication
  • $A\times B\ne B\times A$
    • e.g. let $A$ be size $2\times 3$ and $B$ be size $3\times 2$, $A\times B$ will be size $2\times 2$ but $B\times A$ will be size $3\times 3$
• We can find any value of any term in the resulting matrix using this formula:
  • $c_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+...$

Identities

• Matrices whose size defined by $m\times n$ and $m=n$ are known as square matrices
  • Their size can be denoted with $m\times m$
• These square matrices have a special multiplicative index $I$ for each value of $m$
  • For $2\times 2$ matrices, the identity matrix is $[\genfrac{}{}{0ex}{}{1}{0}\genfrac{}{}{0ex}{}{0}{1}]$
  • An identity matrix is composed of only zeroes except for the diagonal line from the top left corner to the bottom right corner which is filled by ones
    • e.g. a $3\times 3$ identity matrix is $[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}]$
• Identity matrix are special because for a square matrix of the same size, matrix multiplication in any order gives the same square matrix
  • $AI=IA=A$

Matrix Inversion

• Earlier we discussed the existence of identities, however, each square matrix also has an inverse matrix which when they are multiplied together, give the identity matrix
  • The inverse of a matrix $A$ is denoted as the unique matrix $A^{-1}$ with the property that $A{A}^{-1}=I$
    • Inverse matrices are unique, there is only one for a given matrix and it also is the inverse for only one matrix
  • The order for inverse multiplication always gives the same result,
    • $\therefore A{A}^{-1}=A^{-1}A$
• A matrix is invertible if:
  • $A$ has full rank (i.e. rank = #rows)
  • The rows and columns of $A$ are linearly independent

The Inverse of a 2 × 2 Matrix

• If $A=[\genfrac{}{}{0ex}{}{a}{c}\genfrac{}{}{0ex}{}{b}{d}]$, than the inverse of $A$ ($A^{-1}$) is given by:
  • $A^{-1}=\frac{1}{ad-bc}[\genfrac{}{}{0ex}{}{d}{-c}\genfrac{}{}{0ex}{}{-b}{a}]$
    • $ad-bc\ne 0$

Invertible Matrix Theorem

• Go look on slides

The Determinant

• The quantity $ad-bc$ in the inverse formula is called the determinant

  • Notated as $det(A$)

• If $det(A$) = $0$ than $A$ does not have an inverse

• The determinant tells us information about a transformed shape:

  • If the determinant is $a$ ($a\ne 1$), then the transformed shape’s area will be multiplied by $a$
  • If the determinant is negative, then the orientation of the transformed shape will change
    • The transformation involves a reflection

• $\det (A^T)=\det (A)$

• If $A$ has a row of zeroes then $\det (A)=0$

• Read Unit Reader

Computing the determinant of size $n\times n$ requires computing the determinant of $n$ matrices of order $n-1$

Properties of Determinants

• $\det (\alpha A)={\alpha }^n\det (A)$
• $A$ is invertible if and only if $\det (A)\ne 0$

Determinant is Multiplicative

• $\det (AB)=\det (A)\times \det (B)=\det (AB)$
• $\det (A^k)=(\det (A){)}^k$
• $\det (A^{-1})=\frac{1}{\det (A)}$

Determinants of Larger Matrices

• DeepSeek 3x3 Matrix Example:
  
• Comprehensive General Equation: $\det A={\sum }_{j=1}^n(-1{)}^{1+j}{a}_{1j}\det (A[1,j])$
  where $A[i,j]$ is the matrix obtained from $A$ by deleting the $i$th row and $j$th column

Determinants of Large Matrices

• Computing the determinant of one $n\times n$ matrix requires computing the determinant of $n$ matrices of order $n-1$
  • We must use other techniques to compute larger matrices
  • Thus, we use row-reduction to create upper-triangular matrices
  • Finding the determinant of matrices in row-echelon form is easy

Simultaneous Equations Using Matrices

• Cambridge Matrices - 15E
• Simultaneous equations can be written and solved as a matrix multiplication equation
  • Using matrices we can treat the simultaneous equation as one equation and then cancel out matrices on either end until we solve for the desired values
• For example, consider the pair of simultaneous equations:
  • $3x-2y=5,5x-3y=9$
  • This can be written as a matrix equation:
    • $[\genfrac{}{}{0ex}{}{3}{5}\genfrac{}{}{0ex}{}{-2}{-3}][\genfrac{}{}{0ex}{}{x}{y}]=[\genfrac{}{}{0ex}{}{5}{9}]$
  • Let $A$ = $[\genfrac{}{}{0ex}{}{3}{5}\genfrac{}{}{0ex}{}{-2}{-3}]$. The determinant of $A$ is $3(-3)-(-2)5=1$.
  • Since the determinant is non-zero, the inverse matrix exists:
    • $A^{-1}=[\genfrac{}{}{0ex}{}{-3}{-5}\genfrac{}{}{0ex}{}{2}{3}]$
    • Remember that the coefficient for the inverse is not always 1! Use the full formula to find the inverse matrix
  • Now multiply both sides of the original matrix equation by $A^{-1}$:
    • $A^{-1}A[\genfrac{}{}{0ex}{}{x}{y}]=A^{-1}[\genfrac{}{}{0ex}{}{5}{9}]$
    • $I[\genfrac{}{}{0ex}{}{x}{y}]=[\genfrac{}{}{0ex}{}{-3}{-5}\genfrac{}{}{0ex}{}{2}{3}][\genfrac{}{}{0ex}{}{5}{9}]$
      • Since $A^{-1}A=I$
    • $\therefore [\genfrac{}{}{0ex}{}{x}{y}]=[\genfrac{}{}{0ex}{}{3}{2}]$
      • Since $IX=X$
    • $x=3,y=2$
• When the determinant for the given matrix is not $0$, than the equation can be solved by isolating the matrix with unknown variables by multiplying both sides by the inverse of the given matrix
• If the determinant $=0$ than the simultaneous equations have no solution
  • e.g. $x=2y=3,-2x-4y=6$ has no solution as the discriminant of the matrix $[\genfrac{}{}{0ex}{}{1}{-2}\genfrac{}{}{0ex}{}{2}{-4}]$ is equal to 0 and so the matrix has no inverse

Types of Matrices

Transposing Matrices

$A^T$ is the transposed matrix of $A$ and is formed by transposing the rows and columns (flipping the matrix over a diagonal)
e.g. if $A=(\genfrac{}{}{0ex}{}{3}{5}\genfrac{}{}{0ex}{}{-1}{12}\genfrac{}{}{0ex}{}{0}{-4})$ then $A^T=[\begin{array}{cc}3& 5\\ -1& 12\\ 0& -4\end{array}]$

Special Square Matrices

• A matrix is symmetric if $A=A^T$
• A matrix is skew-symmetric if $A=-A^T$
• A matrix is upper-triangular if $A_{ij}=0$ for $i>j$
• A matrix is lower-triangular if $A_{ij}=0$ for $i<j$
• A matrix is diagonal if $A_{ij}=0$ for all $i\ne j$

An $m\times n$ matrix has $m$ rows and $n$ columns

Upper Triangular

• An upper triangular matrix (all numbers below the diagonal are zero) has an easy determinant to compute, equal to the product of all values across diagonal
  • It is easy to solve determinants of matrices in row echelon form

Solving Determinant Using Row-Echelon Form

• When reducing a matrix to solve its determinant, keep track of the changes you make to the original matrix ($A$)
• Thus, when you solve the $\det (A^{\prime })$, you can transform it into $\det (A)$
• The effect of each change is here
  • $\det (A^{\prime })$ does not necessarily equal $\det (A)$!

Subspaces from Matrices

Row and Column Space

• Row space is the span of all rows of a matrix
• Column space is the span of all columns of a matrix
• Null space is all solutions to $Ax=0$ for given matrix $A$
  • To check if a matrix, $B$, is in $A$‘s null space, solve $A\times B$
    • Check if it equals 0

Row and Column Rank

• Row/column/null rank is the dimension of the row/column/null space
  • The dimension of a matrix is equal to the # of non-zero rows/columns in row echelon form
• Row rank is equal to column rank

Rank-Nullity Theorem

If $A$ is an $m\times n$ matrix, then: $rank(A)+nullity(A)=n$
i.e. ‘rank + nullity = number of columns’

Every column is either basic or non-basic:

• Basic $-$ contributes to rank
  • These are columns with leading entries in row echelon form
• Non-basic $-$ contributes to nullity

Explanation of Sowiso Assignment 2 (Q5) in lecture recording