8.3. Matrices to Represent Data http://www.ck12.org
Square matrices have the same number of rows as columns. Theorderof a matrix describes the number of rows
and the number of columns in the matrix. The following matrix is said to have order 2×3 because it has two rows
and three columns. A 1×1 matrix is just a regular number.
[1 2 3
4 5 6]
The identity matrix of ordern×nhas zeros everywhere except along the main diagonal where it has ones. Just like
the number 1 has an important property with numbers, the identity matrix of any order has special properties as well.
[ 1 ],
[1 0
0 1
]
,
1 0 0
0 1 0
0 0 1
When you turn the rows of a matrix into the columns of a new matrix, the two matrices are transpositions of one
another. The superscriptTstands for transpose. Sometimes using the transpose of a matrix is more useful than
using the matrix itself.
A=
[1 2 3
4 5 6
]
AT=
1 4
2 5
3 6
A triangular matrix is not a matrix in the shape of a triangle. Rather, a lower triangular matrix is a square matrix
where every entry below the diagonal is zero. An upper triangular matrix is a square matrix where every entry above
the diagonal is zero. The following is a lower triangular matrix. When you work with solving matrices, look for
triangular matrices because they are much easier to solve.
2 3 4 5
0 4 5 0
0 0 2 − 9
0 0 0 10
A diagonal matrix is both upper and lower triangular which means all the entries except those along the diagonal are
zero. The identity matrix is a special case of a diagonal matrix.
Example A
Organize the driving distances between Sacramento(A), Dallas(B), Albany(C)and Las Vegas(D)in a matrix.
Solution:
A B C D
A
B
C
D
0 1727 2847 560
1727 0 1648 1219
2847 1648 0 2552
560 1219 2552 0
Note that this matrix is symmetric across the main diagonal. Symmetric matrices are important, just like triangular
matrices.
Example B
Kate runs three bakeries and each bakery sells bagels and muffins. The rows represent the bakeries and the columns
represent bagels (left) and muffins (right) sold. Answer the following questions about Kate’s sales.