Chapter 8.4, Problem 93E

To determine

The statement of the square matrix has an entire row of zeros then the determinant of the matrix is zero is true or false.

Answer to Problem 93E

If a square matrix has an entire row or zeroes, the determinant will always is zero is true.

Explanation of Solution

Given information:

The square matrix has an entire row of zeros then the determinant of the matrix is zero

Formula used:

The cofactor expansion is $\left|A\right|=a_{i1}{C}_{i1}+a_{i2}{C}_{i2}+\mathrm{........}+a_{in}{C}_{in}$

Calculation:

If a square matrix has an entire row of zeroes, the determinant will always be zero because if an entire row is row, then each cofactor in the expansion is multiplied by zero.

For example,

  $\left|\begin{array}{ccc}0& 1& 1\\ 0& 0& 2\\ 0& 4& 3\end{array}\right|$

If A is any square matrix, than the determinants of A in terms of cofactor expanding along i^th row is

  $\left|A\right|=a_{i1}{C}_{i1}+a_{i2}{C}_{i2}+\mathrm{........}+a_{in}{C}_{in}$

Or expanding along j^th column is

  $\left|A\right|=a_{1j}{C}_{1j}+a_{2j}{C}_{2j}+\mathrm{........}+a_{nj}{C}_{nj}$

Now, for the given matrix

The determinant of the above matrix by the method of expansion by cofactor,

To expanding from column 2, we need to find $C_{11},C_{21}\text{and}{C}_{31}$

  $\begin{array}{l}{C}_{11}={\left(-1\right)}^{1+1}{M}_{11}\\ =2\\ C_{21}={\left(-1\right)}^{2+1}{M}_{21}\\ =1\\ C_{31}={\left(-1\right)}^{3+1}{M}_{31}\\ =2\end{array}$

Thus we have

  $\begin{array}{l}D=a_{11}{C}_{11}+a_{21}+a_{31}+C_{31}\\ =0\left(2\right)+0\left(1\right)+0\left(2\right)\\ D=0\end{array}$

Since, each cofactor in the expansion is multiplied by zero; determinant of the above matrix is zero.

Therefore, the statement that if a square matrix has an entire row or zeroes, the determinant will always is zero is true.

Conclusion:

If a square matrix has an entire row or zeroes, the determinant will always is zero is true.