If matrix A = [1 2 3] , write AAT.

1.	If matrix A = [1 2 3] , write AAT.
Answer» We are given that, A = [1 2 3] We need to compute AA^T . We know that the transpose of a matrix is a new matrix whose rows are the columns of the original. So, Transpose of matrix A will be given as A^T = \( \begin{bmatrix} 1 \\[0.3em] 2 \\[0.3em] 3 \end{bmatrix}\) Multiplying A by A^T, AA^T = [1 2 3]\( \begin{bmatrix} 1 \\[0.3em] 2 \\[0.3em] 3 \end{bmatrix}\) In multiplication of matrices, [ a₁₁ a₁₂ a₁₃]\( \begin{bmatrix} b_{11} \\[0.3em] b_{21} \\[0.3em] b_{31} \end{bmatrix}\) Dot multiply the matching members of 1^strow of first matrix and 1 st column of second matrix and then sum up. (a₁₁ a₁₂ a₁₃)(b₁₁ b₂₁ b₃₁) = a₁₁ × b₁₁ + a₁₂ × b₂₁ + a₁₃× b₃₁ So, (1 2 3)(1 2 3) = 1 × 1 + 2 × 2 + 3 × 3 ⇒ (1 2 3)(1 2 3) = 1 + 4 + 9 ⇒ (1 2 3)(1 2 3) = 14 Thus, [1 2 3]\( \begin{bmatrix} 1 \\[0.3em] 2 \\[0.3em] 3 \end{bmatrix}\)= [14]

Answer»

We are given that,

A = [1 2 3]

We need to compute AA^T .

We know that the transpose of a matrix is a new matrix whose rows are the columns of the original.

So,

Transpose of matrix A will be given as

A^T = \( \begin{bmatrix} 1 \\[0.3em] 2 \\[0.3em] 3 \end{bmatrix}\)

Multiplying A by A^T,

AA^T = [1 2 3]\( \begin{bmatrix} 1 \\[0.3em] 2 \\[0.3em] 3 \end{bmatrix}\)

In multiplication of matrices,

[ a₁₁ a₁₂ a₁₃]\( \begin{bmatrix} b_{11} \\[0.3em] b_{21} \\[0.3em] b_{31} \end{bmatrix}\)

Dot multiply the matching members of 1^strow of first matrix and 1 st column of second matrix and then sum up.

(a₁₁ a₁₂ a₁₃)(b₁₁ b₂₁ b₃₁) = a₁₁ × b₁₁ + a₁₂ × b₂₁ + a₁₃× b₃₁

So,

(1 2 3)(1 2 3) = 1 × 1 + 2 × 2 + 3 × 3

⇒ (1 2 3)(1 2 3) = 1 + 4 + 9

⇒ (1 2 3)(1 2 3) = 14

Thus,

[1 2 3]\( \begin{bmatrix} 1 \\[0.3em] 2 \\[0.3em] 3 \end{bmatrix}\)= [14]

Discussion