Question

Short answer

The rank of the matrix$\left(\begin{array}{cccc}1& 1& 4& 3\\ 3& 1& 10& 7\\ 4& 2& 14& 10\\ 2& 0& 6& 4\end{array}\right)is2.$

Step-by-step solution

 Step-by-Step Solution Step 1: Given information

The given matrix is.

 Step 2: Definition of the rank of a matrix

The number of non-zero rows remaining in a row reduced matrix is called the rank of a matrix.

Apply row operations to reduce the matrix into row reduction form

Consider the matrix.

Multiply row 1 by 3 and divide row 3 by

Subtract row 1 from row 2 and row 4 from row 3.

Divide row 1 by 3, row 4 by 2 and multiply row 3 by 2.

Subtract row 1 from row 4 and add row 2 to row 3.

Divide row 2 by.

Add row 4 to row 2.

The row reduced form of the given matrix is.

Here, the total number of non-zero rows is 2. Hence, the rank of the matrixis