Matrix is an arrangement of numbers into rows and columns. Make your first introduction with matrices and learn about their dimensions and elements.
A matrix is a rectangular arrangement of numbers into rows and columns.
For example, matrix A has two rows and three columns.

Matrix dimensions

The dimensions of a matrix tells its size: the number of rows and columns of the matrix, in that order.
Since matrix A has start color blueD, t, w, o, space, r, o, w, s, end color blueD and start color goldD, t, h, r, e, e, space, c, o, l, u, m, n, s, end color goldD, we write its dimensions as start color blueD, 2, end color blueD, times, start color goldD, 3, end color goldD, pronounced "two by three".
In contrast, matrix B has start color blueD, t, h, r, e, e, space, r, o, w, s, end color blueD and start color goldD, t, w, o, space, c, o, l, u, m, n, s, end color goldD, so it is a start color blueD, 3, end color blueD, times, start color goldD, 2, end color goldD matrix.
B=[8423121810]B=\left[\begin{array}{rr}-8 & -4 \\ 23 & 12 \\18 &10 \end{array}\right]
When working with matrix dimensions, remember start color blueD, r, o, w, s, end color blueD, times, start color goldD, c, o, l, u, m, n, s, end color goldD!

Check your understanding

1) What are the dimensions of matrix D?
D=[724    2115  11912   0831]D=\left[\begin{array}{rr}-7 & 24 &~~~~2 \\ 1 & 15 &~~11 \\-9 &12&~~~0 \\8 &-3&-1 \end{array}\right]
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
times
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Matrix D has start color blueD, f, o, u, r, space, r, o, w, s, end color blueD and start color goldD, t, h, r, e, e, space, c, o, l, u, m, n, s, end color goldD, so it is a start color blueD, 4, end color blueD, times, start color goldD, 3, end color goldD matrix.
2) What are the dimensions of matrix E?
E=[261308310]E=\left[\begin{array}{rr}-2 & 6 &1 &3 \\ 0& -8 &3&10 \end{array}\right]
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
times
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Matrix E has start color blueD, t, w, o, space, r, o, w, s, end color blueD and start color goldD, f, o, u, r, space, c, o, l, u, m, n, s, end color goldD, so it is a start color blueD, 2, end color blueD, times, start color goldD, 4, end color goldD matrix.
3) What are the dimensions of matrix F?
F=[2010]F=\left[\begin{array}{rr}-2 \\ 0& \\10 \end{array}\right]
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
times
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Matrix F has start color blueD, t, h, r, e, e, space, r, o, w, s, end color blueD and start color goldD, o, n, e, space, c, o, l, u, m, n, end color goldD, so it is a start color blueD, 3, end color blueD, times, start color goldD, 1, end color goldD matrix.

Matrix elements

A matrix element is simply a matrix entry. Each element in a matrix is identified by naming the row and column in which it appears.
For example, consider matrix G:
G=[41471851320422]G=\left[\begin{array}{rr}4& 14&-7 \\ 18 & 5&13 \\-20 &4&22 \end{array}\right]
The element g, start subscript, start color blueD, 2, end color blueD, comma, start color goldD, 1, end color goldD, end subscript is the entry in the start color blueD, s, e, c, o, n, d, space, r, o, w, end color blueD and the start color goldD, f, i, r, s, t, space, c, o, l, u, m, n, end color goldD.
In this case g, start subscript, 2, comma, 1, end subscript, equals, start color maroonD, 18, end color maroonD.
In general, the element in start color blueD, r, o, w, space, i, end color blueD and start color goldD, c, o, l, u, m, n, space, j, end color goldD of matrix A is denoted as a, start subscript, start color blueD, i, end color blueD, comma, start color goldD, j, end color goldD, end subscript.

Check your understanding

4) A=[24   815526   2]A=\left[\begin{array}{rr}2 & -4 &~~~8 \\ 1 & 5&-5 \\-2 &6&~~~2 \end{array}\right]
a, start subscript, 1, comma, 3, end subscript, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

The element a, start subscript, start color blueD, 1, end color blueD, comma, start color goldD, 3, end color goldD, end subscript is the entry in the start color blueD, f, i, r, s, t, space, r, o, w, end color blueD and the start color goldD, t, h, i, r, d, space, c, o, l, u, m, n, end color goldD.
In this case, a, start subscript, 1, comma, 3, end subscript, equals, 8.
5) B=[122.34.61.2]B=\left[\begin{array}{rr}12 & -2.3 \\ 4.6 & 1.2& \end{array}\right]
b, start subscript, 2, comma, 1, end subscript, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

The element b, start subscript, start color blueD, 2, end color blueD, comma, start color goldD, 1, end color goldD, end subscript is the entry in the start color blueD, s, e, c, o, n, d, space, r, o, w, end color blueD and the start color goldD, f, i, r, s, t, space, c, o, l, u, m, n, end color goldD.
In this case, b, start subscript, 2, comma, 1, end subscript, equals, 4, point, 6.
6) Matrix C is a 2, times, 3 matrix with c, start subscript, 1, comma, 2, end subscript, equals, 6.
Which could be matrix C?
Choose 1 answer:
Choose 1 answer:

Since matrix C is a 2, times, 3 matrix, it must have two rows and three columns.
Of the two choices with two rows and three columns, we need to find the one that has an entry of 6 in the first row and second column, since c, start subscript, 1, comma, 2, end subscript, equals, 6.
This corresponds to the following matrix:
[268731]\left[\begin{array}{rr}2 & 6&8 \\7 & -3 &1\end{array}\right]