Matrix Algebra bca 1st wbut

Matrix Algebra
We review here some of the basic denitions and elementary algebraic operations on matrices.
There are many applications as well as much interesting theory revolving around these con-
cepts, which we encourage you to explore after reviewing this tutorial.
A matrix is simply a retangular array of numbers. For example,
A =
266664
a11 a12


a1n
a21 a22


a2n
...
...
. . .
...
am1 am2


amn
377775
is a mn matrix (m rows, n columns), where the entry in the ith row and jth column is aij .
We often write A = [aij ].
Some Terminology
For an n  n square matrix A, the elements a11; a22; : : : ; ann form the main diagonal of
the matrix. The sum
n Pk=1
akk of the elements on the main diagonal of A is called the trace
of A.
The matrix AT = [aji] formed by interchanging the rows and columns of A is called the
transpose of A. If AT = A, the matrix A is symmetric.
Example
Let B = " 6 9
􀀀4 􀀀6 #. The trace of B is 6 + (􀀀6) = 0.
The transpose of B is BT = " 6 􀀀4
9 􀀀6 #.
Addition and Subtraction of Matrices
To add or subtract two matrices of the same size, simply add or subtract corresponding
entries. That is, if B = [bij ] and C = [cij ],
B + C = [bij + cij ] and B 􀀀 C = [bij 􀀀 cij ]: