let a be a 3×2 matrix and b a 2×3 matrix

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let a be a 3×2 matrix and b a 2×3 matrix. show that c = a.b is a singular matrix.

The Correct Answer and Explanation is:

Let matrix AAA be a 3×23 \times 23×2 matrix and matrix BBB be a 2×32 \times 32×3 matrix. Then, the product C=ABC = ABC=AB is defined and results in a 3×33 \times 33×3 matrix.

We aim to show that matrix CCC is singular, meaning it is not invertible or has determinant zero.

Step-by-step Analysis:

  1. Matrix dimensions:
    • AAA: 3 rows, 2 columns ⇒A∈R3×2\Rightarrow A \in \mathbb{R}^{3 \times 2}⇒A∈R3×2
    • BBB: 2 rows, 3 columns ⇒B∈R2×3\Rightarrow B \in \mathbb{R}^{2 \times 3}⇒B∈R2×3
    • C=AB⇒C∈R3×3C = AB \Rightarrow C \in \mathbb{R}^{3 \times 3}C=AB⇒C∈R3×3
  2. Rank consideration:
    • The rank of a matrix product is at most the minimum of the ranks of the two matrices involved.
    • Matrix AAA has at most rank 2 (since it has only 2 columns).
    • Matrix BBB has at most rank 2 (since it has only 2 rows).
    • Therefore, the rank of C=ABC = ABC=AB is at most 2.
  3. Conclusion:
    • Since CCC is a 3×33 \times 33×3 matrix but has rank at most 2, its columns are linearly dependent.
    • This means the determinant of CCC is zero.
    • Hence, matrix CCC is singular.

Explanation

Let us consider a matrix AAA with 3 rows and 2 columns and a matrix BBB with 2 rows and 3 columns. The matrix multiplication ABABAB is defined, and the resulting matrix CCC will have dimensions 3 by 3. However, not every square matrix is invertible. To be invertible, a matrix must have full rank. For a 3×33 \times 33×3 matrix, full rank means rank 3.

Matrix AAA maps vectors from R2\mathbb{R}^2R2 to R3\mathbb{R}^3R3, and matrix BBB maps vectors from R3\mathbb{R}^3R3 to R2\mathbb{R}^2R2. The composite transformation ABABAB thus maps vectors from R3\mathbb{R}^3R3 back into R3\mathbb{R}^3R3 through an intermediate step in R2\mathbb{R}^2R2. However, because the image of BBB lies in a 2-dimensional space, any further mapping through AAA will result in a space of at most dimension 2. Consequently, the product ABABAB cannot span the full R3\mathbb{R}^3R3 space.

This means that the columns of the resulting matrix CCC are linearly dependent, and the determinant of such a matrix must be zero. A zero determinant is the defining property of a singular matrix. Therefore, the product C=ABC = ABC=AB is necessarily a singular matrix due to the rank deficiency inherited from the dimensions of AAA and BBB.

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