Math Problem Statement

diketahui matriks beserta ukurannya yaitu A 3x4 , B 3x4, C 4x2, D 3x2, dan E 4x3 tentukan manakah yang dapat dilakukan, jika tidak dapat dilakukan beri komentar

Diketahui matriks beserta ukurannya yaitu A 3x4 , B 3x4, C 4x2, D 3x2, dan E 4x3 tentukan manakah yang dapat dilakukan, jika tidak dapat dilakukan beri komentar

Solution

Based on the image and the provided matrices:

  • AA is a 3×43 \times 4 matrix.
  • BB is a 3×43 \times 4 matrix.
  • CC is a 4×24 \times 2 matrix.
  • DD is a 3×23 \times 2 matrix.
  • EE is a 4×34 \times 3 matrix.

The expressions given are:

  1. AE+BAE + B
  2. AB+BAB + B
  3. AETAA E^T A
  4. ADT+ETDAD^T + E^T D

Checking Validity:

  1. AE+BAE + B:

    • AEAE: AA is 3×43 \times 4 and EE is 4×34 \times 3, so AEAE results in a 3×33 \times 3 matrix.
    • BB is 3×43 \times 4.
    • The matrices AEAE and BB have different sizes, so the addition cannot be performed.
    • Not possible.
  2. AB+BAB + B:

    • ABAB: AA is 3×43 \times 4 and BB is 3×43 \times 4. Matrix multiplication requires the inner dimensions to match, but they don't (4 ≠ 3).
    • Therefore, ABAB is not defined, and hence the addition is also not possible.
    • Not possible.
  3. AETAA E^T A:

    • ETE^T is the transpose of EE, so ETE^T is 3×43 \times 4.
    • AETAE^T: AA is 3×43 \times 4 and ETE^T is 3×43 \times 4. Their inner dimensions don't match (4 ≠ 3), so AETAE^T is not defined.
    • Therefore, AETAAE^T A is also not possible.
    • Not possible.
  4. ADT+ETDAD^T + E^T D:

    • DTD^T is the transpose of DD, so DTD^T is 2×32 \times 3.
    • ADTAD^T: AA is 3×43 \times 4 and DTD^T is 2×32 \times 3. Their inner dimensions don't match (4 ≠ 2), so ADTAD^T is not defined.
    • Similarly, ETDE^T D: ETE^T is 3×43 \times 4 and DD is 3×23 \times 2. Their inner dimensions don't match (4 ≠ 3), so ETDE^T D is not defined.
    • Hence, the expression ADT+ETDAD^T + E^T D is not possible.
    • Not possible.

Conclusion:

None of the given expressions can be performed due to mismatched matrix dimensions or undefined operations.

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Matrix Addition
Linear Algebra

Formulas

Matrix multiplication: For matrices A (m×n) and B (n×p), the product AB is a matrix of size m×p
Matrix transpose: The transpose of a matrix E (m×n) is a matrix of size n×m

Theorems

Matrix multiplication is defined when the number of columns in the first matrix matches the number of rows in the second matrix

Suitable Grade Level

Grades 11-12