Answer:
The determinant is 1
Explanation:
Given the 3* 3 matrices
, to compute the determinant using the first row means using the row values [0 4 1 ] to compute the determinant. Note that the signs on the values on the first row are +0, -4 and +1
Calculating the determinant;
![= +0\left[\begin{array}{cc}-3&0\\3&1\\\end{array}\right] -4\left[\begin{array}{cc}5&0\\2&1\\\end{array}\right] +1\left[\begin{array}{cc}5&-3\\2&3\\\end{array}\right] \\\\= 0 - 4[5(1)-2(0)] +1[5(3)-2(-3)]\\= 0 -4[5-0]+1[15+6]\\= 0-20+21\\= 1](https://img.qammunity.org/2021/formulas/mathematics/college/d2a5mv3jcdxoiloyhaoh5jcg9mcyeel0yx.png)
The determinant is 1 using the first row as co-factor
Similarly, using the second column
as the cofactor, the determinant will be expressed as shown;
Note that the signs on the values are -4, +(-3) and -3.
Calculating the determinant;
![= -4\left[\begin{array}{cc}5&0\\2&1\\\end{array}\right] -3\left[\begin{array}{cc}0&1\\2&1\\\end{array}\right] -3\left[\begin{array}{cc}0&1\\5&0\\\end{array}\right] \\\\= -4[5(1)-2(0)] - 3[0(1)-2(1)] -3[(0)-5(1)]\\= -4[5-0] -3[0-2]-3[0-5]\\= -20+6+15\\= -20+21\\= 1](https://img.qammunity.org/2021/formulas/mathematics/college/cem9hu42h2twufsi8vtmzxf5cf75ukpajr.png)
The determinant is also 1 using the second column as co factor.
It can be concluded that the same value of the determinant will be arrived at no matter the cofactor we choose to use.