Matrices Presentation by Miss Matkar Pallavi P Matrix

Matrices Presentation by : Miss Matkar Pallavi. P

Matrix - a rectangular array of variables or constants in horizontal rows and vertical columns enclosed in brackets. Element - each value in a matrix; either a number or a constant. Dimension - number of rows by number of columns of a matrix. **A matrix is named by its dimensions.

Examples: Find the dimensions of each matrix. Dimensions: 3 x 2 Dimensions: 4 x 1 Dimensions: 2 x 4

Different types of Matrices • Column Matrix - a matrix with only one column. • Row Matrix - a matrix with only one row. • Square Matrix - a matrix that has the same number of rows and columns.

Equal Matrices - two matrices that have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix. *The definition of equal matrices can be used to find values when elements of the matrices are algebraic expressions.

Examples: Find the values for x and y * Since the matrices are equal, the corresponding elements are equal! * Form two linear equations. * Solve the system using substitution.

* Write as linear equations. * Combine like terms. * Solve using elimination. Now check your answer

3. Set each element equal and solve!

Matrix Multiplication Scalar Multiplication If s is a scalar and A is a matrix 1 4 2 s = 2, and A = 1 s. A = 2 * 4 2 5 3 1 = 2(1) 2(4) 2(2) 2(5) 2(3) 2(1) 2 10 = 8 6 4 2 Rule: if s is a scalar and A is a matrix, then the scalar multiple s. A is the matrix whose columns are s times the corresponding entries in A. In other words, you take the scalar and multiply it with every number in the matrix. Theorem: If r and s are scalars, then (r + s)A = r. A + s. A and r(s. A) = (rs)A.

Matrix Multiplication Before you multiply a matrix by another matrix, you need to check the compatibility of the two matrices. The number of columns (n) in A must equal the number of rows (m) in B in order to carry out the matrix multiplication. Example 1 A= 2 1 5 3 order : B= 3 8 1 2 2× 2 7 4 2× 3 You can see that the number of columns of A matches the number of rows of B. (Later on, you will see that the size of matrix AB will be m of A × n of B). Example 2 A=2 1 5 3 order: 2× 2 B=3 8 4 1 2 1 7 4 2 3× 3 Obviously, n of A ≠ m of B and thus they are not compatible for

Example 3 Find the product AB from the following matrices A= 1 5 0 3 B= 6 4 -7 3 -3 2 -2 -1 A is a 2× 3 matrix and B is a 3× 2 matrix, and they are compatible. To get the final product, we will be sequentially multiplying each row in one matrix by the corresponding column in another matrix. In this example, we take the first row of A and first column of B, multiply the first entries together, second entries together, and third entries together, and then add the three products. 1 0 -2 * 6 = 1(6) + 0(4) + (-2)(-7) = 20 4 -7

Example 3 cont. This sum is one of the entries in the product matrix AB; in fact, being the product of row 1 of A and column 1 of B, it is the (1, 1)-entry in AB. Column 1 ↓ Row 1 → 20 # # # 2 of B. 1 0 Then we continue in like manner; we multiply row 1 of A by column -2 * 3 -3 2 = 1(3) + 0(-3) + (-2)(2) = -1 And its position in matrix AB is Column 2 ↓ 20 -1 ← Row 1 # #

Example 3 cont. Multiplying row 2 of A by column 1 of B gives 5 3 1 * 6 = 5(6)+3(4)+(1)(-7) = 35 4 -7 And its position in matrix AB is at row 2 and column 1. You can do the last multiplication of row 2 of A by column 2 of B by yourself. So the final answer is AB = 1 0 5 3 -2 1 *6 4 -7 3 -3 2 = 20 35 -1 8 Handy trick: As we mentioned before, the size of matrix AB is 2× 2 (A is a 2× 3 matrix and B is a 3× 2 matrix). Also we can see AB ≠ BA

Transpose of a Matrix If we have a given matrix, A that is if the size mxn, then the transpose of it would be an nxm matrix and is denoted by AT. In other words, you take an initial row in the original matrix and make it a column in the transposed matrix, and vice versa with the rows. Let’s look at an example, then AT = A= 5 2 1 , and if we follow the rules

Theorem 1 Let A, B and C be matrices of the same size, and let p and q be scalars. 1. 2. 3. 4. 5. 6. A+B = B+A (A+B) + C = A + (B + C) A+0=A p(A + B) = p. A + p. B (p+q)A = p. A + q. A p(q A) = (pq) A

Theorem 2 If A is an p x q matrix, and of B and C have sizes for which the indicated sums and products are defined, then using some basic laws of arithmetic and some others we can also verify the following statements: 1. A (BC) = (AB) C (associative law of multiplication) 2. A (B + C) = AB + AC (left distributive law) 3. (B+C)A = BA+ CA (right distributive law) r(AB) = (r. A) B = A(r. B) for any scalar r 5. IA = AI (identity for matrix multiplication) 4.

Theorem 3 Let A and B denote matrices whose sizes are compatible enough for the following sums and products. a. (AT )T = A b. ( A+B)T = AT + BT c. For any scalar p, (p A)T = p AT d. ( AB )T = BT AT

Application of Matrix Multiplication The Matrix Application Ax = B To solve system of linear equations

Singular Matrix A square matrix A is said to be singular if l. Al = 0. A is non-singular if IAI = 0

Properties 1. A (adj. A) = |A| In = (adj. A) A 2. If A is a non-singular square matrix of order n, then |adj. A| = |A|n – 1 3. If A and B are non-singular square matrices of same order, then adj AB = (adj. B) (adj. A) 4. If A is a non-singular square matrix, then adj (adj. A) = |A|n– 2 A.

Properties (i) A square matrix is invertible if it is singular. non- (ii) Every invertible matrix possesses a unique inverse. (iii) (AB)– 1 = B– 1 A– 1 or (ABC)– 1 = C– 1 B– 1 A– 1 (iv) (AT)– 1 = (A– 1)T

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