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259 Lecture 14

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259 Lecture 14

Elementary Matrix Theory

A matrix is a rectangular array of elements (usually numbers) written in rows and columns.

Example 1: Some matrices:

- Example 1 (cont.):
- Matrix A is a 3 x 2 matrix of integers.
- A has 3 rows and 2 columns.
- Matrix B is a 2 x 2 matrix of rational numbers.
- Matrix C is a 1 x 4 matrix of real numbers.
- We also call C a row vector.
- A matrix consisting of a single column is often called a column vector.

- Notation:

- Matrices of the same size (i.e. same number of rows and same number of columns), with elements from the same set, can be added or subtracted!
- The way to do this is to add or subtract corresponding entries!

- Example 2: For matrices A and B given below, find A+B and A-B.

- Example 2 (cont):
Solution:

- Note that A+B and A-B are the same size as A and B, namely 2 x 3.

- Matrices can also be multiplied. For AB to make sense, the number of columns in A must equal the number of rows in B.

- Example 3: For matrices A and B given below, find AB and BA.

- Example 3 (cont.):
- A x B is a 3 x 2 matrix. To get the row i, column j entry of this matrix, multiply corresponding entries of row i of A with column j of B and add.
- Since B has 2 columns and A has 3 rows, we cannot find the product BA (# columns of 1st matrix must equal # rows of 2cd matrix).

- Another useful operation with matrices is scalar multiplication, i.e. multiplying a matrix by a number.
- For scalar k and matrix A, kA=Ak is the matrix formed by multiplying every entry of A by k.

- Example 4:

- Recall that for any real number a,
a+0 = 0+a = a and (a)(1) = (1)(a) = a.

- We call 0 the additive identity and 1 the multiplicative identity for the set of real numbers.
- For any real number a, there exists a real number –a, such that
a+(-a) = -a+a = 0.

- Also, for any non-zero real number a, there exists a real number a-1 = 1/a, such that
(a-1)(a) = (a)(a-1) = 1.

- We all –a and a-1 the additive inverse and multiplicative inverse of a, respectively.

- For matrices, we also have an additive identity and multiplicative identity!

A+0 = 0+A = A and AI = IA = A holds.

(HW-check!)

- Clearly, A+(-A) = -A + A = 0 follows! Note also that B-A = B+(-A) holds for any m x n matrices A and B.

- Example 5:

- Example 5 (cont):

- Example 5 (cont.)

- Example 5 (cont.)

- Example 5 (cont):

- For multiplicative inverses, more work is needed.
- For example, here is one way to find the matrix A-1, given matrix A, in the 2 x 2 case!

- From the first matrix equation, we see that e, f, g, and h must satisfy the system of equations:
- ae + bg = 1
af + bh = 0

ce + dg = 0

cf + dh = 1.

- It follows that if e, f, g, and h satisfy this system, then the second matrix equation above also holds!
- Solving the system of equations, we find that ad-bc 0 must hold and
e = d/(ad-bc),

f = -b/(ad-bc),

g = -c/(ad-bc),

h = a/(ad-bc).

- Thus, we have the following result for 2 x 2 matrices:

- In this case, we say A is invertible.
- If ad-bc = 0, A-1 does not exist and we say A is not invertible.
- We call the quantity ad-bc the determinant of matrix A.

- Example 6: For matrices A and B below, find A-1 and B-1, if possible.

- Example 6 (cont.)
- Solution: For matrix A, ad-bc = (1)(4)-(2)(3)= 4-6 = -2 0, so A is invertible. For matrix B, ad-bc = (3)(2)-(1)(6) = 6-6 = 0, so B is not invertible.
- HW-Check that AA-1 = A-1A = I!!
- Note: For any n x n matrix, A-1 exists, provided the determinant of A is non-zero.

- One use of matrices is to solve systems of linear equations.
- Example 7: Solve the system
x + 2y = 1

3x + 4y = -1

- Solution: This system can be written in matrix form AX=b with:

- Example 7 (cont.)
- Since we know from Example 6 that A-1 exists, we can multiply both sides of AX = b by A-1 on the left to get:
A-1AX = A-1b => X = A-1b.

- Thus, we get in this case:

- Example 7 (cont.):

- Elementary Linear Algebra (4th ed) by Howard Anton.
- Cryptological Mathematics by Robert Edward Lewand (section on matrices).