Introduction to Matrices
Systems of Linear Equations
In mathematics, a matrix (plural matrices ) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Matrices are commonly written in box brackets.
An \(m \times n\) (read ‘m by n’) matrix is an arrangement of numbers (or algebraic expressions ) in \(m\) rows and \(n\) columns.
Each number in a given matrix is called an element or entry. A zero matrix has all its elements equal to zero.
Matrices of the same size can be added or subtracted element by element.
\[ A \times B \neq B \times A \]
\[ \left( \begin{array}{cc} a_1 & a_2 \\ b_1 & b_2 \end{array} \right)\left( \begin{array}{cc} c_1 & d_1 \\ c_2 & d_2 \end{array} \right) = \left( \begin{array}{cc} (a_1 \times c_1) + (a_2 \times c_2) & (a_1 \times d_1) + (a_2 \times d_2) \\ (b_1 \times c_1) + (b_2 \times c_2) & (b_1 \times d_1) + (b_2 \times d_2) \end{array} \right) \]
\[ \left( \begin{array}{cc} 1 & 3 \\ 0 & 2 \end{array} \right)\left( \begin{array}{cc} 1 & 2 \\ 4 & 1 \end{array} \right) = \left( \begin{array}{cc} (1 \times 1) + (3 \times 4) & (1 \times 2) + (3 \times 1) \\ (0 \times 4) + (2 \times 4) & (0 \times 2) + (2 \times 1) \end{array} \right) = \left( \begin{array}{cc} 14 & 5 \\ 8 & 2 \end{array} \right) \]
The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus and more.
The symbol for determinant is two vertical lines either side: |A| means the determinant of the matrix A (exactly the same symbol as absolute value.)
In the case of a 2 × 2 matrix the determinant can be defined as
\[{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}}\] Similarly, for a 3 × 3 matrix A, its determinant is
\[{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}}\]
Consider the following matrix \(A\) \[A = \right(\begin{array}{ccc} 6 & 3 & 2 \\ 4 & 7 & 5 \\ \end{array}\left) \]
A square matrix is a matrix with the same number of rows and columns. An \(n\times n\) matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied.
If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix.
The identity matrix or unit matrix of size n is the \(n\times n\) square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by \(I_n\), or simply by I if the size is immaterial or can be trivially determined by the context.
\[ I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},\ \cdots ,\ I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}} \]
When A is a square matrix (i.e. \(n\times n\)), it is a property of matrix multiplication that \[I_{N}A=AI_{n}=A.\,\]
When A is \(m\times n\), it is a property of matrix multiplication that \[I_{m}A=AI_{n}=A.\,\]
The Identity matrix itself is invertible, being its own inverse.
The Identity matrix is symmetrix, which is to say, it is it’s own tranpose.
\[ \right(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\left) \]
A symmetric matrix is a matrix that is identical to it’s tranpose.
Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics.
Specifically, a matrix is in row echelon form if
\[\left[ \begin{array}{ccccc} 1 & a_0 & a_1 & a_2 & a_3 \\ 0 & 0 & 1 & a_4 & a_5 \\ 0 & 0 & 0 & 1 & a_6 \end{array} \right]\]
\[ {\begin{array}{ccccc} 1&2 & 6 &3 & 5 \\ 0&1&4 &0 & 6 \\ 0&1&2 &1 & 7 \\ 0&0&1 &1 & 7 \end{array}} \]
\[ {\begin{array}{ccccc} 1&1 & 6 &8 & 5 \\ 0&2&6 &0 & 6 \\ 0&0&1 &1 & 7 \\ 0&0&0 &1 & 7 \end{array}} \]
\[ {\begin{array}{ccccc} 1&1 & 6 &8 & 5 \\ 0&1&6 &0 & 6 \\ 0&0&0 &1 & 7 \\ 0&0&1 &1 & 7 \end{array}} \]
[ { \[\begin{array}{ccccc} 1&1 & 6 &8 & 5 \\ 0&1&6 &0 & 6 \\ 0&0&0 &0 & 1 \end{array}\]} ] }
\[ A = PDP^{-1} \]
By extension \[ D = P^{-1}AP\]
Given the matrices \(A\) and \(B\), where
\[A = \begin{bmatrix} 1 & 3 & 2 \\ 2 & 0 & 1 \\ 5 & 2 & 2 \end{bmatrix} , \quad B = \begin{bmatrix} 4 \\ 3 \\ 1 \end{bmatrix},\]
the augmented matrix (\(A|B\)) is written as
\[(A|B)= \left[\begin{array}{ccc|c} 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & 3 \\ 5 & 2 & 2 & 1 \end{array}\right].\]
\[ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\\end{bmatrix} = \begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \\ \end{bmatrix}\]
\[ a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} = b_1 \\ a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} = b_2 \\ a_{31}x_{1} + a_{32}x_{2} + a_{33}x_{3} = b_3 \\ \]
Consider the following matrices \(A\) and \(B\) which are given as
\[ A = \left( \begin{array}{ccc} 0 & 2 & 1 \\ 2 &1 &2\\ 1 &2 &0\\ \end{array} \right) \qquad B = \left( \begin{array}{ccc} 1 &0 &1\\ 2 &1 &0\\ 1 &2 &1\\ \end{array} \right) \]
Show that \(AB\) and \(BA\) are not equal.
Find matrices \(C\), \(D\) and \(E\) such that
\(B\) + \(C\) = \(A\)
\(BD\) = \(B\)
\(B\) - \(E\) = \(A\).
\[A = \begin{bmatrix} -4 & 3 \\ 2 & 1 \\ \end{bmatrix} , \qquad B = \begin{bmatrix} 4 & -2 \\ 1 & 3 \\ \end{bmatrix}, \qquad C = \begin{bmatrix} -4 & 0 & 5 \\ 1 & -3 & 2\\ \end{bmatrix} \]
Calculate \(2BC\)
Calculate \((A+B)C\)
\[A = \begin{bmatrix} 1 & 3 \\ -2 & 4 \\ \end{bmatrix} , \qquad B = \begin{bmatrix} 2 & -3 & 7 \\ 1 & 0 & 2 \\ \end{bmatrix}, \qquad C = \begin{bmatrix} 1 & - 2 \\ 4 & 3 \\ 0 & 5 \\ \end{bmatrix} \]
Write down the augmented matrix for the following system of equations.
Use Gaussian elimination to solve the system.
Calculate the following if possible: \(AB\),\(BA\), \(AC\) and \(C^{T}\) \[A = \begin{bmatrix} 2 & 2 \\ 3 & 6 \\ \end{bmatrix} , \qquad B = \begin{bmatrix} -1 & 3 \\ 2 & -2 \\ \end{bmatrix}, \qquad C = \begin{bmatrix} 2 & - 2 \\ 1 & 4 \\ 1 & 1 \\ \end{bmatrix} \]
Given the matrices
\[A = \begin{bmatrix} 0 & 2 & 3 \\ 2 & 3 & 1 2 & 1 & 3 \\ \end{bmatrix} \mbox{ and } B = \begin{bmatrix} 2 \\ 4 \\ 2 \\ \end{bmatrix} \]
Calculate \(AB\).
Find a matrix \(X\) such that \(AX = B\)
Find a matrix \(Y\) such that \(AY = A\)
For the vectors given below, evaluate the following expressions where it is possible. \[ \vec{u}=\left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right], \vec{v}=\left[ \begin{array}{c} -1 \\ 0 \\ 4 \end{array}\right], \vec{x}=\left[ \begin{array}{c} 3 \\ 4 \end{array}\right], \vec{y}=\left[ \begin{array}{c} -4 \\ 3 \end{array}\right], \vec{w}=\left[ \begin{array}{c} 1 \\ 0\\ 2 \\ -5 \end{array}\right], \vec{z}=\left[ \begin{array}{c} 2 \\ 2 \\ 2 \\ 3 \end{array}\right] \]
Calculate the angles between the pairs \(\vec{u},\vec{v}\), \(\vec{x},\vec{y}\), and \(\vec{w},\vec{z}\) from the previous question. Give your answers in both radians and degrees.
For the matrices below, evaluate the following expressions where it is possible. \[ A=\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right], B=\left[ \begin{array}{cc} -2 & 0 \\ 1 & -7 \end{array}\right], C=\left[ \begin{array}{ccc} 3 & 2 & -2 \\ 4 & 8 & 2 \end{array}\right], D=\left[ \begin{array}{ccc} 3 & 2 & -2 \\ 4 & 8 & 2 \end{array}\right], \] \[ E=\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right], F=\left[ \begin{array}{ccc} -1 & 0 & 2 \\ 3 & 4 & 1 \\ 3 & 1 & 0 \end{array}\right], \]
\[ G=\left[ \begin{array}{cc}3 & 4 \\1 & 2 \\2 &-1 \\ \end{array}\right], H=\left[ \begin{array}{ccc}3 & 4 & 3\\1 & 2 & 2\\ \end{array}\right], I=\left[ \begin{array}{ccc}2 & 2 & 1\\ \end{array}\right], \]
\[ J=\left[ \begin{array}{c} 3\\ 1\\ 1\\ \end{array}\right], K=\left[ \begin{array}{ccc} 2 & 1 & 3\\1 & 2 & 2\\2 & 1 & 0\\\end{array}\right], \]
\[2x + y - z = 2\] \[x - y + z = 4\] \[x + 2y + 2z = 10\]