MAT 322: LINEAR ALGEBRA

MAT 322: LINEAR ALGEBRA

Introduction The study of Linear Algebra essentially begins with the Mathematical array of numbers called MATRIX. The design of this scheme assumes that students have good knowledge of matrices and matrix algebra. It is therefore necessary to assist the students by first giving a quick revision of matrix and matrix algebra.

A matrix is an array of numbers Denoted with a Capital letter Every matrix has an order (or dimension): that is, the number of rows  the number of columns. So, A is 2 by 3 or (2  3).

MATRIX EQUALITY Two matrices are equal if and only if; they both have the same number of rows and the same number of columns their corresponding elements are equal SQUARE MATRIX A square matrix is a matrix that has the same number of rows and columns (n  n)

DIAGONAL MATRIX A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero. IDENTITY MATRIX An identity matrix is a diagonal matrix where the diagonal elements all equal one.

A single row or column of numbers denoted with bold small letters Null Matrix A square matrix where all elements equal zero. VECTOR A single row or column of numbers denoted with bold small letters row vector column vector

MAIN (PRINCIPAL) DIAGONAL The elements a11, a22, a33, MAIN (PRINCIPAL) DIAGONAL The elements a11, a22, a33, ... constitute the main or principal diagonal of the matrix A = [aij], if it is square. Eg. TRIANGULAR MATRIX A matrix in which all the entries below or above the main diagonal are zeros are called upper and lower triangular matrices respectively. If only the main diagonal is non-zero, then it is simply triangular matrix

Matrix Operations Transposition Addition and Subtraction Multiplication Inversion

The transpose of A is denoted by A' or (AT). If TRANSPOSE OF A MATRIX The transpose of a matrix is a new matrix that is formed by interchanging the rows as columns. The transpose of A is denoted by A' or (AT). If EXAMPLE Given that A = then A‘ = Notice that the first and the last elements are always the same.

Addition and Subtraction Two matrices may be added (or subtracted) iff they are the same order. Simply add (or subtract) the corresponding elements. So, A + B = C yields. Eg. where

MATRIX SCALAR MULTIPLICATION To multiply a scalar and a matrix, simply multiply each element of the matrix by the scalar quantity e.g. If k is a scalar and

MATRIX MULTIPLICATION In order to multiply two matrices say AB, they must be CONFORMABLE that is, the number of columns in A must equal the number of rows in B. So, A  B = C (m  n)  (n  p) = (m  p) (m  n)  (p  n) = cannot be done (1  n)  (n  1) = a scalar (1x1)

Matrix Multiplication (cont.) Thus where

Matrix Multiplication- an example Thus where,

AB does not necessarily equal BA (BA may even be an impossible operation) Eg., A  B = C (2  3)  (3  2) = (2  2) B  A = D (3  2)  (2  3) = (3  3) Matrix multiplication is Associative A(BC) = (AB)C Multiplication and transposition (AB)' = B'A'

Note that a row matrix multiplied by a column matrix when conformable gives a scalar

The rank of a matrix is defined as rank(A) = number of linearly independent rows = the number of linearly independent columns. A set of vectors is said to be linearly dependent if scalars c1, c2, …, cn (not all zero) can be found such that: c1a1 + c2a2 + … + cnan = 0 Example: a = [1 21 12] and b = [1/3 7 4] are linearly dependent A matrix A of dimension n  p (p < n) is of rank p. Then A has maximum possible rank and is said to be of full rank. In general, the maximum possible rank of an n  p matrix A is min(n,p). TRACE The trace of a matrix is the sum of the main diagonal elements of the a square matrix

The Inverse of a Matrix (A-1) For an n  n matrix A, there may be a B such that AB = I = BA. The inverse is comparable to a reciprocal A matrix which has an inverse is nonsingular. A matrix which does not have an inverse is singular. An inverse exists only if PROPERTIES OF INVERSE

CALCULATION OF INVERSE OF A MATRIX If and |A|  0 ADJOINT To get the adjoint of a matrix; Substitute each element by its cofactor taking note of the signs. Transpose the resulting matrix in (i) above.

Cofactors The minor of each element is called a cofactor if a +ve or –ve sign is assigned to it. The sign is determined by the position of the element. If the sum i+j of the element is even, the sign is +ve. It is –ve if i+j of the element is odd.

EXAMPLE

Ex 2: (a) Find the adjoint matrix of A (b) Use the adjoint matrix of A to find A–1 Sol:

cofactor matrix of A adjoint matrix of A inverse matrix of A ※ The computational effort of this method to derive the inverse of a matrix is high (especially to compute the cofactor matrix for a higher-order square matrix) ※ However, for computers, it is easy. since it is not necessary to judge which row operation should be used and the only thing needed to do is to calculate determinants of matrices Check:

Echelon Matrix When the number of zeros preceding the first non-zero entry of a row increases row by row until only zero rows remain; that is if there exists non-zero entries, such a matrix is called echelon matrix or is in echelon form

DISTINGUISHED ELEMENTS ECHELON MATRICES DISTINGUISHED ELEMENTS The first non-zero in each row. Thus in eg.1 above 1 and 4 are the distinguished elements and in eg.2 the DE are 1,1,1 in eg.3 the DE are 2,7,6). The 3rd example above is an example of a row reduced echelon matrix. The zero matrix irrespective of the number of rows or columns is a row echelon matrix.

ELEMENTARY ROW OPERATIONS Two matrices A and B are said to be row equivalence if B can be obtained from A by a finite sequence of any or all the following operations called Elementary Row Operation; Interchange the i-th row and the j-th row Multiply the i-th row by a non zero scalar say k Replace the i-th row by k times the j-th row plus the i-th row Replace the i-th row by k’ times the j-th row plus k (non-zero) times the i-th row.

EXAMPLES

EXERCISES Transform to Echelon form Transform to row reduced form

DETERMINANT OF A MATRIX The determinant of a matrix A is denoted by |A| (or det(A)). Determinants exist only for square matrices. DETERMINANT OF 2X2 MATRIX Examples

Ex 3: The determinant of a square matrix of order 3 Sol:

Subtract these three products Alternative way to calculate the determinant of a square matrix of order 3: Subtract these three products Add these three products

–4 6 16 Ex: Recalculate the determinant of the square matrix A in Ex 3 6 16 ※ This method is only valid for matrices of order 3

Ex 4: The determinant of a square matrix of order 4

Sol: ※ By comparing Ex 4 with Ex 3, it is apparent that the computational effort for the determinant of 4×4 matrices is much higher than that of 3×3 matrices. In the next section, we will learn a more efficient way to calculate the determinant

Determinant of a Triangular Matrix If A is an n  n triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is ※ Based on the above, it is straightforward to obtain that ※ On the next slide, only take the case of upper triangular matrices for example to prove . It is straightforward to apply the following proof for the cases of lower triangular and diagonal matrices

Ex 6: Find the determinants of the following triangular matrices (b) Sol: (a) |A| = (2)(–2)(1)(3) = –12 (b) |B| = (–1)(3)(2)(4)(–2) = 48

3.2 Evaluation of a Determinant Using Elementary Row Operations The computational effort to calculate the determinant of a square matrix with a large number of n is unacceptable. In this section, we show how to reduce the computational effort by using elementary operations Note: Elementary row operations and determinants Let A and B be square matrices Notes: The above three properties remains valid if elementary column operations are performed to derive column-equivalent matrices (This result will be used in Ex 5 on Slide 3.25)

Ex: (check the characteristics of determinants

NOTE: Conditions that yield a zero determinant If A is a square matrix and any of the following conditions is true, then det(A) = 0 (a) An entire row (or an entire column) consists of zeros (Perform the cofactor expansion along the zero row or column) (b) Two rows (or two columns) are equal (c) One row (or column) is a multiple of another row (or column) (For (b) and (c), based on the mathematical induction , perform the cofactor expansion along any row or column other than these two rows or columns) Notes: For conditions (b) or (c), you can also use elementary row or column operations to create an entire row or column of zeros and obtain the results. ※ Thus, we can conclude that a square matrix has a determinant of zero if and only if it is row- (or column-) equivalent to a matrix that has at least one row (or column) consisting entirely of zeros

Ex:

PROPERTIES OF DETERMINATES Determinants have several mathematical properties which are useful in matrix manipulations. |A|=|A'|. If a row or column of A = 0, then |A|= 0. If every value in a row or column is multiplied by k, then |A| = k|A|. If two rows (or columns) are interchanged the sign, but not value, of |A| changes. If two rows or columns are identical, |A| = 0. If two rows or columns are linear combination of each other, |A| = 0 |A| remains unchanged if each element of a row or each element multiplied by a constant, is added to any other row. 8. |AB| = |A| |B| 9. Det of a diagonal matrix = product of the diagonal elements

Determinant of a matrix product det(AB) = det(A) det(B) (Verified by Ex 1 on the next slide) Notes: (2) (3) (4) (There is an example to verify this property on Slide 3.33) (Note that this property is also valid for all rows or columns other than the second row)

Ex 1: The determinant of a matrix product Find |A|, |B|, and |AB| Sol:

Check: |AB| = |A| |B|

Ex: Pf:

Ex 2: Theorem 3.6: Determinant of a scalar multiple of a matrix If A is an n × n matrix and c is a scalar, then det(cA) = cn det(A) (can be proven by repeatedly use the fact that ) Ex 2: Sol:

(Determinant of an invertible matrix) A square matrix A is invertible (nonsingular) if and only if det(A)  0 If A is invertible, then AA–1 = I. , we can have |A||A–1|=|I|. Since |I|=1, neither |A| nor |A–1| is zero Suppose |A| is nonzero. It is aimed to prove A is invertible. By the Gauss-Jordan elimination, we can always find a matrix B, in reduced row-echelon form, that is row-equivalent to A 1. Either B has at least one row with entire zeros, then |B|=0 and thus |A|=0 since |Ek|…|E2||E1||A|=|B|. →← 2. Or B=I, then A is row-equivalent to I, and by Theorem 2.15 (Slide 2.59), it can be concluded that A is invertible

Ex 3: Classifying square matrices as singular or nonsingular Sol: A has no inverse (it is singular) B has inverse (it is invertible/nonsingular)

Ex 4: Determinant of an inverse matrix Determinant of a transpose (a) (Based on the mathematical induction , compare the cofactor expansion along the row of A and the cofactor expansion along the column of AT) Ex 4: (a) (b) Sol:

(2) Ax = b has a unique solution for every n × 1 matrix b Equivalent conditions for a nonsingular matrix: If A is an n × n matrix, then the following statements are equivalent (1) A is invertible (2) Ax = b has a unique solution for every n × 1 matrix b (3) Ax = 0 has only the trivial solution (4) A is row-equivalent to In (5) A can be written as the product of elementary matrices (6) det(A)  0 ※ The statements (1)-(5) are collected in Theorem 2.15, and the statement (6) is from Theorem 3.7

Ex 5: Which of the following system has a unique solution? (b)

Sol: (a) This system does not have a unique solution (b) This system has a unique solution

MATRICES AND SYSTEMS OF LINEAR EQUATIONS (CRAMERS’ RULE)

Use Cramer’s rule to solve the system of linear equation

CAYLEY-HAMILTON THEOREM

CHECK SLIDE 90 END

END

END

END

END

END

FINAL TEST QUESTIONS Answer all questions. Time 30mins