Question

Give an example of a\(3 \times 3\)matrix\(A\)with all nonzero entries such that\(\det A = 13\).

Short answer

Therefore, the example of\(3 \times 3\)matrix is given by,

\(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&{13}\\2&1&3\end{aligned}} \right)\).

Step-by-step solution

Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangularaligned.

The numbers are called the elements, or entries, of the matrix.

If there are\(m\) rows and \(n\) columns, the matrix is said to be a “\(m\) by\(n\)” matrix, written “\(m \times n\).”

To find \(3 \times 3\)matrix

We want to find a\(3 \times 3\) matrix,

\(A = \left( {\begin{aligned}{*{20}{l}}a&b&c\\d&e&f\\g&h&i\end{aligned}} \right)\), with all non-zero entries, such that:

\(aei + bfg + cdh - ceg - bdi - afh = 13\).

For\(a = b = c = 1\), we have

\(ei + fg + dh - eg - di - fh = 13\).

For\(f = 13,g = 2,h = 1\), we're left with:

\(ei + d - 2e - di = 0\).

For\(d = 1,e = 2,i = 3\), the equation applies.

So one example of such a matrix is\(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&{13}\\2&1&3\end{aligned}} \right)\).