Science:Math Exam Resources/Courses/MATH221/December 2009/Question 02 (b)/Solution 1

Strategy:

Use the following determinant properties.

The determinant of upper triangular matrix (i.e a matrix in row echelon form in this case) is equal to the product of its diagonal entries. If i$\neq$j , then subtracting some constant time row i from row j does not change the determinant.

Step 1:

Perform the following row operations on B:

row 2 = row 2 - 2 row 1

row 3 = row 3 - 2 row 1

and we get

${\begin{bmatrix}1&0&3&0\\0&1&-2&-1\\0&2&-5&0\\0&3&-1&1\end{bmatrix}}$

Since the above matrix is still not in upper triangular form, perform the following row operations to it:

row 3 = row 3 - 2 row 2

row 4 = row 4 - 3 row 2

Then we get:

${\begin{bmatrix}1&0&3&0\\0&1&-2&0\\0&0&-1&2\\0&0&5&4\end{bmatrix}}$

After performing one more row operation,

row 4 = row 4 + 5 row 3,

we get the following upper triangular matrix:

${\begin{bmatrix}1&0&3&0\\0&1&-2&0\\0&0&-1&0\\0&0&0&14\end{bmatrix}}$

Step 2 :

Let $U'={\begin{bmatrix}1&0&3&0\\0&1&-2&0\\0&0&-1&0\\0&0&0&14\end{bmatrix}}$

Then $det(A)=det(U')=(1)(1)(-1)(14)=-14$ and thus $det(A)=-14$.