Math 141: Business Mathematics I
Fall 2015
§2.4 Matrices
Instructor: Yeong-Chyuan Chung
Outline
• Equality of matrices
• Matrix operations - addition/subtraction, transpose, scalar multiplication
Matrices
A matrix is simply a rectangular array of numbers. In §2.2 and §2.3, we used (augmented)
matrices to represent systems of linear equations as part of the Gauss-Jordan elimination
method for solving systems of linear equations. Matrices can also be used to represent data
in an organized manner, just like tables but without headings and lines separating rows and
columns. If data is represented in this way, then sometimes we can perform operations on
matrices to get more information about the data. We’ll see more of this in §2.5.
If a matrix has m rows and n columns, then we say that it has size m × n (read “m by n”),
or we say that it is an m × n matrix.
If we name a matrix as A, then we sometimes usethe notation
aij to indicate the entry in
2 3
the ith row and jth column. For example, if A =
, then a21 = 5.
5 6
Equality of Matrices
Two matrices are equal if they have the same size and their corresponding entries are equal.
2 3
2 3 4
For example,
6=
because they have different sizes.
4 6
4 6 7
2 3
2 4
Also,
6=
because they differ in one entry although they have the same size.
4 6
4 6
1
§2.4 Matrices
2
Matrix Operations
Addition/Subtraction
Two matrices can be added or subtracted only if they have the same size. In this case, we
just perform the operation entrywise.
1 2
2 4 6
Example.
+
doesn’t make sense since the two matrices have different
3 4
3 5 7
sizes.
1 2
6 1
1−6 2−1
−5 1
−
=
=
3 4
2 5
3−2 4−5
1 −1
Addition of matrices follows rules similar to those for addition of numbers. For example,
A + B = B + A, and (A + B) + C = A + (B + C).
Transpose
When we turn the rows of a matrix into columns, and turn the columns into rows, we get
a new matrix called the transpose of the original matrix. More precisely, we obtain the
transpose of a matrix by writing the first row as the first column, writing the second row as
the second column, and so on. If we call the original matrix A, then we write AT for the
transpose of A.


1 20
1 3 5
Example. If A =
, then AT = 3 40.
20 40 60
5 60
The transpose operation has the following properties:
• If A is an m × n matrix, then AT is an n × m matrix.
• (AT )T = A.
Scalar multiplication
The word “scalar” here refers to a real number. Scalar multiplication means multiplying
a scalar to a matrix. This is done simply by multiplying the scalar to every entry of the
matrix. If A is a matrix and c is a real number, then we write cA for the matrix obtained
by multiplying c to A.
3 5
6 10
Example. If A =
, then 2A =
.
7 9
14 18
§2.4 Matrices
3
Example (Exercise 24 in the text). Solve for u, x, y, and z in the given matrix equation.
x −2
−2 z
4 −2
+
=
3 y
−1 2
2u 4
Example (Exercise 26 in the text). Solve for u, x, y, and z in the given matrix equation.




T
y−1
2
−4 −u
1 3 x
2  = 2  0 −1 
− 3 1
2 4 −1
4
2z + 1
4
4
4
§2.4 Matrices
−2 1
2 −3
Example (Exercise 28 in the text). Let A =
and B =
. Find a matrix
0 3
1 −2
X satisfying the matrix equation 3X − A + 2B = 0.