MAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2.

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1 MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1 Review: Linear Algebra II Vectors and matrices 1 11 Definition 1 12 Operations 1 2 Linear Algebra III Inverses and Determinants 1 21 Inverse Matrices 1 22 Determinants Determinants for matrices of size Determinants for matrices of size Solving system of linear equations using inverse matrices 6 11 Definition 1 Review: Linear Algebra II Vectors and matrices 12 Operations Basic matrix opertaions Matrix-Vector multiplication Matrix-Matrix multiplication 21 Inverse Matrices Example 1 2 Linear Algebra III Inverses and Determinants Definition Suppose that A = a ij is an n n square matrix If there exists an n n square matrix B such that AB = BA = I n then B is called the inverse matrix of A and denoted by A 1 Note: If the matrix A has an inverse, A is called invertible or nonsingular If A does not have inverse, then A is called singular If A is invertible, its inverse matrix is unique; that is, if B and C are both inverse matrices of A, then B = C Date:

2 2 JING LI If A an invertible n n matrix, then (A 1 ) 1 = A If A and B are invertible n n matrices, then (AB) 1 = B 1 A 1 Question 1 for this subsection: and what the inverse is? Example 2 To find the inverse matrix of A = Solutions: How can we find out whether a given matrix has an inverse Algorithm to compute an inverse matrix: Given a square matrix A, we take the identity matrix I and write the system A I Then we use the three elementary row operations to reduce the left hand side to the identity matrix If this is possible, we end up with I B = I A 1 If we cannot reduce the left hand side to the identity matrix, then the original matrix A is not invertible Example 3 Find the inverse of A =

3 Example 4 Find the inverse of A = MAT 1332: CALCULUS FOR LIFE SCIENCES Example 5 Find the inverse of A = Example 6 Find the inverse of A =

4 4 JING LI Question 2 for this subsection: When is a matrix invertible? Sometimes we are not interested in the exact inverse of a matrix but only in the question of whether or not the matrix is invertible a11 a Example 7 For a 2 2 matrix A = 12, when is this matrix invertible? a 21 a 22 The solution to the example above will lead the next section 22 Determinants In algebra, the determinant is a special number associated with any square matrix From Example 7 of 2 2 matrix, we know that if a 11 a 22 a 12 a 22 = 0, then A cannot be invertible Since the expression a 11 a 22 a 12 a 21 is so important, we give it a special name 221 Determinants for matrices of size 2 2 The determinant of 2 2 matrices will be discussed in this subsection Definition For a 2 2 matrix The determinant of A is given by A = If det(a) 0 then A is invertible and a11 a 12, a 21 a 22 det(a) = a 11 a 22 a 12 a 21 A 1 = 1 det(a) If det(a) = 0 then A is not invertible a22 a 12 a 21 a 11 Example 8 Example 3 revisited

5 MAT 1332: CALCULUS FOR LIFE SCIENCES 5 Example 9 Example 4 revisited Example 10 Calculate the determinate of A = the inverse matrix , and then use the formula to calculate Example 11 Calculate the determinate of A = the inverse matrix , and then use the formula to calculate Example 12 Use the determinate to determine whether the matrix A = is invertible 222 Determinants for matrices of size 3 3 Determinants can be defined for square matrices of all sizes, and it is always true that if det(a) 0 then the matrix A is invertible We will only consider the case of 3 3-matrices here, since it has a fairly simple form Determinants of bigger matrices can be computed, but it takes time

6 6 JING LI Definition The determinant of the matrix is given by A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 det(a) = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 31 a 22 a 13 a 32 a 23 a 11 a 33 a 21 a 12, Note: There is no really simple formula for the inverse of a 3 3 matrix (or larger) analogous to the one for 2 2 case To find the inverse, we still have to use the row-reduction algorithm How to memorize the formula for the determinant of 3 3 matrix? This determinant formula can be remembered more easily by attaching the first two columns of the matrix A as columns 4 and 5 of a larger matrix and then taking products along the diagonal down with plus signs and products along the diagonal up with minus signs, ie, Example 13 Example 5, revisited a 11 a 12 a 13 a 11 a 12 a 21 a 22 a 23 a 21 a 22 a 31 a 32 a 33 a 31 a 32 Example 14 Example 6, revisited 23 Solving system of linear equations using inverse matrices To finish this lecture, let us recall that a system of linear equations a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a n1 x 1 + a n2 x a nn x n = b n

7 MAT 1332: CALCULUS FOR LIFE SCIENCES 7 which can be rewritten as (1) Ax = b, where, A: coefficient matrix of the linear system; x = (the right side of equations) If A is invertible, then multiplying (1) by A 1 from the left, we find that Since A 1 A = I n and I n x = x, it follows that A 1 Ax = A 1 b x = A 1 b This can be summarized into the following box: x 1 x 2 x n (unknown variables); b = b 1 b 2 b n Important Observation: If A is an invertible n n square matrix, then the unique solution of the system of linear equations Ax = b is given by x = A 1 b Let s look at the following example, Example 15 Solve the following system { 2x1 + x 2 = 7 3x 1 2x 2 = 14 which has the matrix in Example 10 as the coefficient matrix