Concept explainers
Rotations If a point
a. If the point
b. Multiplication by what matrix would result in a counterclockwise rotation of 90°? 135°? (Express the matrices in terms of R.) [HinT: Think of a rotation through 90° as two successive rotations through 45°.]
c. Multiplication by what matrix would result in a clockwise rotation of 45°?
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Chapter 4 Solutions
Finite Mathematics
- Guided proof Prove the associative property of matrix addition: A+B+C=A+B+C. Getting Started: To prove that A+B+C and A+B+C are equal, show that their corresponding entries are equal. i Begin your proof by letting A, B, and C be mn matrices. ii Observe that the ij th entry of B+C is bij+cij. iii Furthermore, the ij th entry of A+B+C is aij+bij+cij. iv Determine the ij th entry of A+B+C.arrow_forwardTrue or false? det(A) is defined only for a square matrix A.arrow_forwardGuided Proof Prove that the inverse of a symmetric non-singular matrix is symmetric. Getting Started: To prove that the inverse of A is symmetric, you need to show that A-1T=A-1. i Let A be symmetric, nonsingular matrix. ii This means that AT=A and A-1 exists. iii Use the properties of the transpose to show that A-1T is equal to A-1.arrow_forward
- Guided Proof Prove that the determinant of an invertible matrix A is equal to 1 when all of the entries of A and A1 is integers. Getting Started: Denote det(A) as x and det(A1) as y. Note that x and y are real numbers. To prove that det(A) is equal to 1, you must show that both x and y are integers such that their product xy is equal to 1. (i) Use the property for the determinant of a matrix product to show that xy=1. (ii) Use the definition of a determinant and the fact that the entries of A and A1 are integers to show that both x=det(A) and y=det(A1) are integers. (iii) Conclude that x=det(A) must be either 1 or 1 because these are the only integer solutions to the equation xy=1arrow_forwardFinding a Value: Find x such that the matrix is equal to its own inverse. A=3x23arrow_forwardGuided Proof Prove that if A is row-equivalent to B, and B is row-equivalent to C, A is row-equivalent to C. Getting Started: to prove that If A is row-equivalent to C, you have to find elementary matrices E1, E2. Ek such that A=EkE2E1C. i Begin by observing that A is row-equivalent to B and B is row-equivalent to C. ii This means that there exist elementary matrices F1F2Fn and G1G2Gm such that A=FnF2F1B and B=GmG2G1C. iii Combine the matrix equations from step ii.arrow_forward
- Prove part b of Theorem 1.35. Theorem 1.35 Special Properties of Let be an arbitrary matrix over. With as defined in the preceding paragraph,arrow_forwardGuided Proof Prove Property 2 of Theorem 3.3: When B is obtained from A by adding a multiple of a row of A to another row of A, detB = detA. Getting Started: To prove that the determinant of B is equal to the determinant of A, you need to show that their respective cofactor expansions are equal. i Begin by letting B be the matrix obtained by adding c times the jth row of A to the ith row of A. ii Find the determinant of B by expanding in this ith row. iii Distribute and then group the terms containing a coefficient of c and those not containing a coefficient of c. iv Show that the sum of the terms not containing a coefficient of c is the determinant of A, and the sum of the terms containing a coefficient of c is equal to 0.arrow_forward
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