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If A and B are 5 x 3 matrices, and Cl is a 2 x 5 matrix, which of the following are defined? ✔A. CB B. CB - A C. C+B ✓D.A - B E. CT ✔ F. BAT G. AB

If A and B are 5 x 3 matrices, and Cl is a 2 x 5 matrix, which of the following are defined? ✔A. CB B. CB - A C. C+B ✓D.A - B E. CT ✔ F. BAT G. AB

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Matrix Operation Definitions** The problem statement: If \( A \) and \( B \) are \( 5 \times 3 \) matrices, and \( C \) is a \( 2 \times 5 \) matrix, which of the following operations are defined? **Options:** - A. \( CB \) ✅ - B. \( CB - A \) - C. \( C + B \) - D. \( A - B \) ✅ - E. \( C^T \) - F. \( BA^T \) ✅ - G. \( AB \) **Explanation:** 1. **\( CB \):** For matrix multiplication \( CB \), matrix \( C \) (\( 2 \times 5 \)) is multiplied by matrix \( B \) (\( 5 \times 3 \)), resulting in a \( 2 \times 3 \) matrix. This is defined. 2. **\( CB - A \):** This operation requires both \( CB \) and \( A \) to be of the same dimensions for subtraction. Since \( CB \) is \( 2 \times 3 \) and \( A \) is \( 5 \times 3 \), this is not defined. 3. **\( C + B \):** Addition of matrices requires matching dimensions, but \( C \) is \( 2 \times 5 \) and \( B \) is \( 5 \times 3 \). Not defined. 4. **\( A - B \):** Both matrices \( A \) and \( B \) have the dimensions \( 5 \times 3 \), so their subtraction is defined. 5. **\( C^T \):** The transpose of \( C \) would be a \( 5 \times 2 \) matrix. 6. **\( BA^T \):** For multiplication, \( B \) (\( 5 \times 3 \)) and \( A^T \) (\( 3 \times 5 \)) result in a \( 5 \times 5 \) matrix. This is defined. 7. **\( AB \):** Matrix multiplication requires the inner dimensions to match, but \( A \) (\( 5 \times 3 \)) and \( B \) (\( 5 \times 3 \)) do not satisfy this condition
Transcribed Image Text:**Matrix Operation Definitions** The problem statement: If \( A \) and \( B \) are \( 5 \times 3 \) matrices, and \( C \) is a \( 2 \times 5 \) matrix, which of the following operations are defined? **Options:** - A. \( CB \) ✅ - B. \( CB - A \) - C. \( C + B \) - D. \( A - B \) ✅ - E. \( C^T \) - F. \( BA^T \) ✅ - G. \( AB \) **Explanation:** 1. **\( CB \):** For matrix multiplication \( CB \), matrix \( C \) (\( 2 \times 5 \)) is multiplied by matrix \( B \) (\( 5 \times 3 \)), resulting in a \( 2 \times 3 \) matrix. This is defined. 2. **\( CB - A \):** This operation requires both \( CB \) and \( A \) to be of the same dimensions for subtraction. Since \( CB \) is \( 2 \times 3 \) and \( A \) is \( 5 \times 3 \), this is not defined. 3. **\( C + B \):** Addition of matrices requires matching dimensions, but \( C \) is \( 2 \times 5 \) and \( B \) is \( 5 \times 3 \). Not defined. 4. **\( A - B \):** Both matrices \( A \) and \( B \) have the dimensions \( 5 \times 3 \), so their subtraction is defined. 5. **\( C^T \):** The transpose of \( C \) would be a \( 5 \times 2 \) matrix. 6. **\( BA^T \):** For multiplication, \( B \) (\( 5 \times 3 \)) and \( A^T \) (\( 3 \times 5 \)) result in a \( 5 \times 5 \) matrix. This is defined. 7. **\( AB \):** Matrix multiplication requires the inner dimensions to match, but \( A \) (\( 5 \times 3 \)) and \( B \) (\( 5 \times 3 \)) do not satisfy this condition
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Step 1: Concept used in the solution
  • The product AB is possible if the number of columns of A is equal to the number of rows of B.
  • A+B and A-B are possible if the order of A is equal to the order of B.
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