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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:. Matrix Definition. Matrix Definition. Example 1 (cont.): Matrix A is a 3 x 2 matrix of integers.

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259 Lecture 14

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259 lecture 14

259 Lecture 14

Elementary Matrix Theory


Matrix definition

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

Example 1: Some matrices:

Matrix Definition


Matrix definition1

Matrix Definition

  • 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.


Matrix definition2

Matrix Definition

  • Notation:


Arithmetic with matrices

Arithmetic with Matrices

  • 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!


Arithmetic with matrices1

Arithmetic with Matrices


Arithmetic with matrices2

Arithmetic with Matrices

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


Arithmetic with matrices3

Arithmetic with Matrices

  • Example 2 (cont):

    Solution:

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


Arithmetic with matrices4

Arithmetic with Matrices

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


Arithmetic with matrices5

Arithmetic with Matrices

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


Arithmetic with matrices6

Arithmetic with Matrices

  • 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).


Arithmetic with matrices7

Arithmetic with Matrices

  • 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.


Arithmetic with matrices8

Arithmetic with Matrices

  • Example 4:


Identities and inverses

Identities and Inverses

  • 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.


Identities and inverses1

Identities and Inverses

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


Identities and inverses2

Identities and Inverses

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

(HW-check!)


Identities and inverses3

Identities and Inverses

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


Identities and inverses4

Identities and Inverses

  • Example 5:


Identities and inverses5

Identities and Inverses

  • Example 5 (cont):


Identities and inverses6

Identities and Inverses

  • Example 5 (cont.)


Identities and inverses7

Identities and Inverses

  • Example 5 (cont.)


Identities and inverses8

Identities and Inverses

  • Example 5 (cont):


Identities and inverses9

Identities and Inverses

  • 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!


Identities and inverses10

Identities and Inverses


Identities and inverses11

Identities and Inverses

  • 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:


Identities and inverses12

Identities and Inverses

  • 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.


Identities and inverses13

Identities and Inverses

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


Identities and inverses14

Identities and Inverses

  • 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.


Linear systems of equations

Linear Systems of Equations

  • 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:


Linear systems of equations1

Linear Systems of Equations

  • 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:


Linear systems of equations2

Linear Systems of Equations

  • Example 7 (cont.):


References

References

  • Elementary Linear Algebra (4th ed) by Howard Anton.

  • Cryptological Mathematics by Robert Edward Lewand (section on matrices).


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