Game Development Reference
In-Depth Information
As an example, find the product:
4
1
1
3
AB
2
1
2
1
We verify the product is defined because the number of columns of
A
equals the number of rows of
B
. Also note that the resulting matrix is a
2
2 matrix. Using formula (4), we have:
4
1
1
3
a
b
a
b
4
1
1
2
4
1
3
1
6
13
1
1
1
2
AB
2
1
2
1
a
b
a
b
2
1
1
2
2
1
3
1
0
5
2
1
2
2
As an exercise, verify for this particular example that
AB
BA
.
A more general example:
b
b
11
12
a
a
a
a
b
a
b
a
b
a
b
a
b
a
b
11
12
13
11
11
12
21
13
31
11
12
12
22
13
32
AB
b
b
C
21
22
a
a
a
a
b
a
b
a
b
a
b
a
b
a
b
21
22
23
21
11
22
21
23
31
21
12
22
22
23
32
b
b
31
32
The Identity Matrix
There is a special matrix called the
identity matrix
. The identity matrix
is a square matrix that has zeros for all elements except along the main
diagonal, and the elements along the main diagonal are all ones. For
example, below are 2
2, 3
3, and 4
4 identity matrices:
1
0
0
1
0
1
0
0
0
0
1
0
0
1
0
0
0
1
0
0
1
0
0
1
0
0
0
0
1
The identity matrix acts as a multiplicative identity:
MI=IM=M
That is, multiplying a matrix by the identity does not change the
matrix. Further, multiplying with the identity matrix is a particular case
when matrix multiplication is commutative. The identity matrix can be
thought of as the number “1” for matrices.
Example
: Verify that multiplying the matrix
M
12
04
by the 2
2
identity matrix results in
M
.
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