Lesson Objectives
  • Learn the basic definition of a matrix
  • Learn about naming matrices and matrix notation
  • Learn how to determine the order of a matrix
  • Learn how to determine if two matrices are equal

Understanding Matrix Fundamentals


We previously learned how to solve a system of linear equations using matrix methods. We saw that a matrix is an array of numbers that is arranged in rows (horizontal) and columns (vertical), and placed inside of square brackets (in some cases, we use parentheses). An example of a typical matrix with the rows and columns labeled Although we commonly work with numbers, we can also see variables, symbols, or a combination of numbers, variables, and symbols as the entries in our matrix. We can use any capital letter such as A to name a matrix. $$ A = \left[ \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{array} \right] $$

Elements of a Matrix

Each individual entry in a matrix is commonly referred to as an entry or element of the matrix. The subscript notation is used to name the individual elements of a matrix. Notation such as aij refers to an element from matrix A located in row i and column j. With subscript notation, the row is always first, followed by the column second. If we think about matrix A above, when we see a11 (you can read this as "a sub one one") this refers to the element or entry in the first row and first column. Similarly, a21 would refer to the element in the second row and first column. In some cases, you will see commas introduced into the subscript notation as a way to separate the row number from the column number. This is especially useful when multi-digit numbers get involved for the row or column. Notation such as a11,23 refers to the element in row 11 and column 23. Let's look at an example.
Example #1: Find a13 and a32. $$ A = \left[ \begin{array}{ccc} 4 & -7 & 9 \\ -11 & 5 & 12 \\ 8 & 1 & -2 \\ -3 & 7 & -8 \\ \end{array} \right] $$ $$a_{13} = 9$$ This represents the element in the first row and third column. $$a_{32} = 1$$ This represents the element in the third row and second column.

The Order of a Matrix

The order of a matrix is given as m × n, where m is the number of rows and n is the number of columns. Note: we may also refer to the order as the dimensions or the size of the matrix.
  • 3 × 2 matrix has 3 rows and 2 columns
  • 2 × 4 matrix has 2 rows and 4 columns
  • 5 × 3 matrix has 5 rows and 3 columns
Let's look at an example.
Example 2: Find the order of the matrix. $$ A = \left[ \begin{array}{cc} 1 & -4 \\ 22 & -3 \\ 18 & 11 \\ \end{array} \right] $$ Since the matrix has 3 rows and 2 columns, the order is 3 × 2.

Special Names

In some cases, matrices are given special names.
  • A rectangular matrix has a different number of rows and columns
    • You may see this as an m × n matrix, m ≠ n
    • Ex: 4 × 3 matrix is a rectangular matrix
  • A square matrix has the same number of rows as columns
    • You may see this as an n × n matrix
    • Ex: 3 × 3 matrix is a square matrix
  • A matrix with one row is a row matrix
    • Ex: 1 × 3 matrix is a row matrix
  • A matrix with one column is a column matrix
    • Ex: 3 × 1 matrix is a column matrix
There will be other matrices with special names that will be introduced as we continue through this chapter. Let's look at an example.
Example #3: Give the special name for each matrix below. $$ A = \left[ \begin{array}{cc} -1 & 15 \\ 9 & 18 \\ \end{array} \right] $$ Since matrix A is a 2 × 2, it is considered a square matrix. It has the same number of rows (2) as columns (2). $$ B = \left[ \begin{array}{cccc} 19 & 13 & -7 & -5 \\ \end{array} \right] $$ Since matrix B is a 1 × 4, it is considered a row matrix. $$ C = \left[ \begin{array}{c} -8 \\ -3 \\ 17 \\ 1 \\ -4 \end{array} \right] $$ Since matrix C is a 5 × 1, it is considered a column matrix. $$ D = \left[ \begin{array}{cc} 9 & 7 & -8 \\ 10 & 11 & -4 \\ \end{array} \right] $$ Since matrix D is a 2 × 3, it is considered a rectangular matrix.

Equality of Matrices

Two matrices are equal if they have the same order and if all corresponding elements are equal. Let's look at some examples.
Example #4: Determine if matrix A is equal to matrix B. $$ A = \left[ \begin{array}{cc} 3 & -8 \\ 5 & 7 \\ \end{array} \right] $$ $$ B = \left[ \begin{array}{cc} -2 & -8 \\ 5 & 7 \\ \end{array} \right] $$ The two matrices A and B are not equal (A ≠ B). Although they have the same order (both are 2 × 2), not all corresponding elements are equal. $$a_{11} = 3$$ $$b_{11} = -2$$ Matrices A and B are not equal for the element in the first row and first column (position (1,1)).
Example #5: Find the values for the variables for which A = B. $$ A = \left[ \begin{array}{ccc} 1 & -2 & x\\ 9 & 8 & -3\\ \end{array} \right] $$ $$ B = \left[ \begin{array}{ccc} 1 & -2 & 5\\ 9 & y & -3\\ \end{array} \right] $$ Since we want A = B, this means the order must be the same and corresponding elements must be equal.
  • Each matrix is a 2 × 3
  • All corresponding elements without variables are equal
To complete the problem we need to look at the corresponding elements where variables are involved. $$a_{13} = x \, \text{and} \, b_{13} = 5$$ $$x = 5$$ $$a_{22} = 8 \, \text{and} \, b_{22} = y$$ $$y = 8$$

Skills Check:

Example #1

Find the order of the matrix. $$\left[ \begin{array}{ccc}3&9&2\\ 5&6&4 \\2& 1& 9\end{array}\right]$$

Please choose the best answer.

A
9 × 3
B
3 × 6
C
3 × 5
D
4 × 3
E
3 × 3

Example #2

Find the indicated element. $$A = \left[ \begin{array}{cccc}5&2&1&7\\ -4&-8&6&10\end{array}\right]$$ $$a_{24} = \, ?$$

Please choose the best answer.

A
$$a_{24} = 10$$
B
$$a_{24} = 6$$
C
$$a_{24} = 7$$
D
$$a_{24} = 5$$
E
$$a_{24} = -4$$

Example #3

What value of x makes A = B? $$ A = \left[ \begin{array}{cccc} 1 & -2 & x & -9\\ 8 & -6 & -1 & 41\\ 11 & 14 & 23 & -4 \\ \end{array} \right] $$ $$ B = \left[ \begin{array}{cccc} 1 & -2 & 19 & -9\\ 8 & -6 & -1 & 41\\ 11 & 14 & 23 & -4 \\ \end{array} \right] $$

Please choose the best answer.

A
$$x = -1$$
B
$$x = 23$$
C
$$x = 19$$
D
$$x = 14$$
E
$$x = -6$$
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