Question

Find the transpose of the matrix. D=[[4,-7,18],[-7,0,25],[18,25,-36]]

Answer 1

To find the transpose of matrix D, each row of the original matrix is rewritten as a column. The transpose of matrix D is actually the same as the original matrix since D is symmetric. The transpose is a new matrix that mirrors the original across its diagonal.

Step-by-step explanation:

To find the transpose of a matrix, you simply rewrite the rows as columns. Here's the matrix D being transposed step-by-step:

Original matrix D:
\[D=\begin{bmatrix} 4 & -7 & 18 \\ -7 & 0 & 25 \\ 18 & 25 & -36 \end{bmatrix}\]

Transpose of D, denoted as \(D^T\), is calculated as:

1. Take the first row (4, -7, 18) and make it the first column of \(D^T\).
2. Take the second row (-7, 0, 25) and make it the second column of \(D^T\).
3. Take the third row (18, 25, -36) and make it the third column of \(D^T\).

1. This results in the transposed matrix \(D^T\):
2. 
   
3. \[D^T=\begin{bmatrix} 4 & -7 & 18 \\ -7 & 0 & 25 \\ 18 & 25 & -36 \end{bmatrix}\]

Notice that the transpose of matrix D is actually the same as the original matrix because D is a symmetric matrix (its transpose is equal to the original matrix).