Concept explainers
Guide Proof Prove that if
Getting Started: To prove that the matrices
(i) Begin your proof by letting
(ii) The
(iii) Evaluate the entries
(iv) Repeat this analysis for the product
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Chapter 2 Solutions
Bundle: Elementary Linear Algebra, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term
- 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_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_forwardProof Let A and B be nn matrices such that AB=I.Prove that |A|0 and |B|0.arrow_forward
- A square matrix is called upper triangular if all of the entries below the main diagonal are zero. Thus, the form of an upper triangular matrix is where the entries marked * are arbitrary. A more formal definition of such a matrix . 29. Prove that the product of two upper triangular matrices is upper triangular.arrow_forwardTrue or false? det(A) is defined only for a square matrix A.arrow_forwardWhich of the following operations can we perform for a matrix A of any dimension? (i) A+A (ii) 2A (iii) AAarrow_forward
- Proof Prove that if A is an nn matrix, then A-AT is skew-symmetric.arrow_forwardSimilar Matrices In Exercises 19-22, use the matrix P to show that the matrices A and Aare similar. P=A=A=[11201]arrow_forwardGuidedProof Prove that if A is an mn matrix, then AAT and ATA are symmetric matrices. Getting Started: To prove that AAT is symmetric, you need to show that it is equal to its transpose, AATT=AAT. i Begin your proof with the left-hand matrix expression AATT. ii Use the properties of the transpose operation to show that AATT can be simplified to equal the right-hand expression, AAT. iii Repeat this analysis for the product ATA.arrow_forward
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