Matrices

Last updated Jun 16, 2023 Edit Source

# ← Specialist Home

• 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 MatricesCircle

• A matrix is a rectangular array of mathematical objects (usually numbers)
  • E.g. $A = ({3\atop5}{-1\atop12}{0\atop-4})$
    • Matrices are represented with capital letters, such as $A$
  • If it has $m$ rows and $n$ columns, it is called a $m\\ ×\\ n$ matrix
    • The matrix above is a $2\\ ×\\ 3$ 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

• Two matrices can be added or subtracted by adding or subtracting the corresponding entries
  • Example:
    • If $A = ({1\atop-2}{6\atop5})$ and $B = ({2\atop3}{-8\atop0})$
    • Then $A + B = ({1\\ +\\ 2\atop-2\\ +\\ 3}{6\\ +\\ (-8)\atop5\\ +\\ 0}) = ({3\atop1}{-2\atop5})$
  • Matrices can only be added/subtracted if they are of the same size
    • $m_1 = m_2,\\ n_1 = n_2$

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

• To multiply by a scalar, multiply each entry by the scalar
  • Scalars are real numbers
• Example:
  • If $A = ({1\atop-2}{6\atop5})$, then $3A = ({3\\ ×\\ 1\atop3\\ ×\\ -2}{3\\ ×\\ 6\atop3\\ ×\\ 5}) = ({3\atop-6}{18\atop15})$

• Matrix multiplication is more complex than the other topics we have seem so far
  • Example of matrix multiplication:
    • Let $A = [{1\atop4}{3\atop2}]$ and $B = [{5\atop6}{1\atop0}]$
    • $A \\ ×\\ B = AB = [{1\atop4}{3\atop2}][{5\atop6}{1\atop0}]$
    • $[{1\\ ×\\ 5\\ +\\ 3\\ ×\\ 6\atop4\\ ×\\ 5\\ +\\ 2\\ ×\\ 6}\\ {1\\ ×\\ 1\\ +\\ 3\\ ×\\ 0\atop4\\ ×\\ 1\\ +\\ 2\\ ×\\ 0}]$
    • $[{23\atop32}{1\atop4}]$
  • 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
    • $a_{11}\\ ×\\ b_{11},\\ a_{12}\\ ×\\ b_{21}$
    • These values are added together to get the value in matrix $C$
• $A$ has dimensions $m\\ ×\\ n$ and $B$ has dimensions $p\\ ×\\ q$, if $n = p$ they can be multiplied
  • The resultant matrix is $m\\ ×\\ q$ in size
    • e.g. a matrix of size $2\\ ×\\ 3$ multiplied by a matrix $3\\ ×\\ 4$ will have a size of $2\\ ×\\ 4$, 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
  • $A\\ ×\\ B \neq B\\ ×\\ A$
    • e.g. let $A$ be size $2\\ ×\\ 3$ and $B$ be size $3\\ ×\\ 2$, $A\\ ×\\ B$ will be size $2\\ ×\\ 2$ but $B\\ ×\\ A$ will be size $3\\ ×\\ 3$
• We can find any value of any term in the resulting matrix using this formula:
  • $c_{ij} = a_{i1}\\ b_{1j}\\ +\\ a_{i2}\\ b_{2j}\\ +\\ …$

• Matrices whose size defined by $m\\ ×\\ n$ and $m = n$ are known as square matrices
  • Their size can be denoted with $m\\ ×\\ m$
• These square matrices have a special multiplicative index $I$ for each value of $m$
  • For $2\\ ×\\ 2$ matrices, the identity matrix is $[{1\atop0}{0\atop1}]$
  • 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 $3\\ ×\\ 3$ identity matrix is $[\\ \begin{smallmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{smallmatrix}\\ ]$
• Identity matrix are special because for a square matrix of the same size, matrix multiplication in any order gives the same square matrix
  • $AI = IA = A$

• 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 $AA^{-1} = I$
    • 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,
    • $\therefore AA^{-1} = A^{-1}A$

• # The Inverse of a $2\\ ×\\ 2$ Matrix:

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

• # The Determinant

  • The quantity $ad - bc$ in the inverse formula has a special name, the determinant
    • Notated as $det(A$)
  • If $det(A$) = $0$ than $A$ does not have an inverse

• 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:
    • $[{3 \atop 5}{-2 \atop -3}][{x \atop y}] = [{5 \atop 9}]$
  • Let $A$ = $[{3 \atop 5}{-2 \atop -3}]$. The determinant of $A$ is $3(-3) - (-2)5 = 1$.
  • Since the determinant is non-zero, the inverse matrix exists:
    • $A^{-1} = [{-3 \atop -5}{2 \atop 3}]$
    • 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 $A^{-1}$:
    • $A^{-1}A[{x \atop y}] = A^{-1}[{5 \atop 9}]$
    • $I[{x \atop y}] = [{-3 \atop -5}{2 \atop 3}][{5 \atop 9}]$
      • Since $A^{-1}A = I$
    • $\therefore [{x \atop y}] = [{3 \atop 2}]$
      • 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 $[{1 \atop -2}{2 \atop -4}]$ is equal to 0 and so the matrix has no inverse