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Difficulty: EasyMatrices and Determinants

If matrix AA has dimension 2×32 \times 3 and matrix BB has dimension 3×43 \times 4, what is the dimension of the matrix product ABAB?

  1. A
    3×33 \times 3
  2. 2×42 \times 4Answer
  3. C
    4×24 \times 2
  4. D
    2×32 \times 3

Answer

The dimension of the matrix product ABAB is 2×42 \times 4.
For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. When a matrix of size m×nm \times n is multiplied by a matrix of size n×pn \times p, the resulting product matrix has size m×pm \times p. For A2×3A_{2 \times 3} and B3×4B_{3 \times 4}, the outer dimensions give 2×42 \times 4.

Step-by-Step Solution

1
Identify the dimensions of the given matrices AA and BB
Matrix AA is of order 2×32 \times 3 (m=2,n=3m = 2, n = 3) and Matrix BB is of order 3×43 \times 4 (n=3,p=4n = 3, p = 4).
Matrix multiplication requirement requires the number of columns of the first matrix to match the number of rows of the second matrix.
2
Apply the matrix multiplication dimension rule
The product matrix ABAB has mm rows and pp columns, resulting in dimension 2×42 \times 4.
When multiplying an m×nm \times n matrix by an n×pn \times p matrix, the resulting matrix has dimensions m×pm \times p.

Key Concept

Matrix Multiplication Dimension Compatibility and Resulting Order
Estimated Time:45s
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Topics

Matrices and Determinants

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