If \(\rm A=\begin{matrix} 2&3&1\\1&4&1 \end{matrix}\) and \(\rm B=\begin{matrix} 6&1\\0&3\\5&2 \end{matrix} \) find ATBT

Explanation:

Concept:

Transpose of Matrix:

• The transpose of a matrix is found by interchanging its rows into columns or columns into rows.
• The transpose of the matrix is denoted by using the letter “T” in the superscript of the given matrix. For example, if “A” is the given matrix, then the transpose of the matrix is represented by A’ or A^T.
• If A = [a_ij]_m*n  then, A^T = [aij]_n*m

Matrix multiplication:

• Matrix multiplication, also known as matrix product and the multiplication of two matrices, produces a single matrix.
• It is a type of binary operation. If A and B are the two matrices, then the product of the two matrices A and B are denoted by: X = AB
• To perform the multiplication of two matrices, one should make sure that the number of columns in the 1^st matrix is equal to the rows in the 2^nd matrix. Therefore, the resulting matrix product will have a number of rows of the 1^st matrix and a number of columns of the 2^nd matrix.
• Consider the matrix A which is the m × n matrix and matrix B, which is the n × p matrix. Then, matrix C = AB is defined as the A × B matrix.
• An element in matrix C, C_xy is defined as: 

          \(C_{xy} = \sum_{k=1}^{n} A_{xk}B_{ky}\)                    

          ; For x = 1 to m and y= 1 to p.

If \(\rm A=\begin{bmatrix} a&b\\c&d \end{bmatrix}\) and \(\rm B=\begin{bmatrix} x&y\\z&w \end{bmatrix}\)

Then, X = A × B =\(\begin{bmatrix} a \times x+b \times z&a \times y+b \times w\\c \times x+d \times z&c \times y+d \times w \end{bmatrix}\)

Calculation:

Data given,

\(\rm A=\begin{bmatrix} 2&3&1\\1&4&1 \end{bmatrix}\) , \(\rm B=\begin{bmatrix} 6&1\\0&3\\5&2 \end{bmatrix} \)

Now, A is a 2 × 3 matrix and B is a 3 × 2 matrix.

So, AT will be a 3 × 2 matrix and BT will be a 2 × 3 matrix.

\(\rm A^T=\begin{bmatrix} 2&1\\3&4\\1&1 \end{bmatrix} \) and \(\rm B^T=\begin{bmatrix} 6&0&5\\1&3&2 \end{bmatrix}\)

So, matrix multiplication is possible and the output matrix will be a 3 × 3 matrix. 

\(\rm A^TB^T=\begin{bmatrix} 2×6+1×1&2×0+1×3&2×5+1×2\\3×6+4×1&3×0+4×3&3×5+4×2\\1×6+1×1&1×0+1×3&1×5+1×2 \end{bmatrix} \)

∴ \(A^TB^T=\rm \begin{bmatrix} 13&3&12\\22&12&23\\7&3&7 \end{bmatrix}\)