Determinants (Practice Question)

A determinant is a single numerical value calculated from a square matrix. It helps determine properties such as whether a matrix has an inverse and whether a system of linear equations has a unique solution.

Solved Examples

Example 1: If x, y, and z are different. and A = \begin{vmatrix} \begin{vmatrix} x & x^{2} & 1 + x^{3} \\ y & y^{2} & 1 + y^{3} \\ z & z^{2} & 1 + z^{3} \end{vmatrix} = 0 , then show that 1 + xyz = 0.

Solution:

Using Sum Property

\begin{vmatrix} x & x^{2} & 1 + x^{3} \\ y & y^{2} & 1 + y^{3} \\ z & z^{2} & 1 + z^{3} \end{vmatrix} = \begin{vmatrix} x & x^{2} & 1 \\ y & y^{2} & 1\\ z & z^{2} & 1 \end{vmatrix}  + \begin{vmatrix} x & x^{2} & x^{3} \\ y & y^{2} & y^{3} \\ z & z^{2} & z^{3} \end{vmatrix} \text{} \\ = (-1)^{2}\begin{vmatrix} 1 & x & x^{2} \\ 1 & y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} + xyz\begin{vmatrix} 1 & x & x^{2} \\ 1 & y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} \\ = (1 + xyz) \begin{vmatrix} 1 & x & x^{2} \\ 1 & y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} = 0

On solving this determinant and expanding it,

A = (1 + xyz)(y- x)(z-y)(z-x)
Since it's given in the question, that all x, y and z have different values and A =0. So the only term that can be zero is 1 + xyz.
Hence, 1 + xyz = 0

Example 2: Evaluate the \begin{vmatrix} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \\ \end{vmatrix} .

Solution:

Using Scalar Multiple Property and Repetition Property

\begin{vmatrix} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \\ \end{vmatrix} = \begin{vmatrix} 17(6) & 6(3) & 6(6) \\ 1 & 3 & 4 \\ 17 & 3 & 6 \\ \end{vmatrix} \\ = 6\begin{vmatrix} 17 & 3 & 6 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \\ \end{vmatrix} = 0  \text{}

Example 3: Evaluate the determinant A = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 5 \\ 2 & 3 & 1 \end{vmatrix} \\

Solution:

Using Proportionality Property
Two of the rows of the matrix are identical.

So, A = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 5 \\ 2 & 3 & 1 \end{vmatrix} \\ = 0 \text{}

Example 4: Given Matrix: A = \begin{bmatrix}3 & 5 & 2 \\0 & 4 & 1 \\0 & 0 & 6\end{bmatrix}.

Solution:

Given the matrix:
A = \begin{bmatrix}3 & 5 & 2 \\0 & 4 & 1 \\0 & 0 & 6\end{bmatrix}
Identify the Diagonal Elements
a₁₁ = 3, a₂₂ = 4, a₃₃ = 6
Apply the Determinant Formula for Triangular Matrices
det(A) = a₁₁ × a₂₂ × a₃₃
det(A) = 3× 4 × 6 = 72

Example 5: Verify det(Δ') = k det(Δ) in,

Δ = \begin{vmatrix} 5 & 2 & 3\\ 2 & 4 & 5\\ 1 & 8 & 7 \end{vmatrix} k = 3/2

Solution:

Δ = \begin{vmatrix} 5 & 2 & 3\\ 2 & 4 & 5\\ 1 & 8 & 7 \end{vmatrix}
det(Δ) = 5[(4×7) - (8×5)] - 2[(2×7) - (5×1)] + 3[(2×8) - (4×1)]
det(Δ) = -60 - 18 + 36
det(Δ) = -42

Now, on multiplying by k = 3/2, first column,

Δ' = \begin{vmatrix} 5 & 2 & 3\\ 2 & 4 & 5\\ 1 & 8 & 7 \end{vmatrix}
Δ' = \begin{vmatrix} 15/2 & 2 & 3\\ 3 & 4 & 5\\ 3/2 & 8 & 7 \end{vmatrix}
det (Δ') = -63

Therefore,
det(Δ') = 3/2  det(Δ)

⇒ det(Δ') = k det(Δ)

Example 6: Find the Determinant of

[A] = \begin{pmatrix}9 & 8 & 7\\ 0 & 0 & 0\\1 & 8 & 5\\\end{pmatrix}

Solution:

|A| = \begin{vmatrix}9 & 8 & 7\\ 0 & 0 & 0\\1 & 8 & 5\\\end{vmatrix}
det(Δ) = 0 (since R₂ ⇢0)

Example 7: Find the Determinant of

[A] = \begin{pmatrix}2 & 0 & 0 & 0\\ 3 & 1 & 0 & 0\\5 & 6 & 8 & 0\\7 & 1 & 5 & 9\end{pmatrix}

Solution:

|A| = \begin{vmatrix}2 & 0 & 0 & 0\\ 3 & 1 & 0 & 0\\5 & 6 & 8 & 0\\7 & 1 & 5 & 9\end{vmatrix}
det(Δ) = 2 × 1 × 8 × 9
det(Δ) = 14

Practice Problems on Properties of Determinants

Question 1: Calculate the determinant of the following matrix and state how row swapping affects the determinant:

\begin{vmatrix}2 & 3 \\1 & 4\end{vmatrix}

Swap the rows and then calculate the determinant again.

Question 2: Determine the determinant of the matrix below. Then, multiply the first row by 3 and find the new determinant:

\begin{vmatrix}1 & -1 \\2 & 3\end{vmatrix}

Compare the original determinant with the new one to explain the effect of scalar multiplication on the determinant.

Question 3: Calculate the determinant before and after performing a row operation where you add twice the first row to the second row:

\begin{vmatrix}1 & 2 \\3 & 4\end{vmatrix}

Question 4: Calculate the determinant of the following triangular matrix:

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

Discuss why the determinant of a triangular matrix is the product of its diagonal elements.

Question 5: Find the determinant of the matrix below, noting what happens when rows are proportional:

\begin{vmatrix}3 & 6 \\6 & 12\end{vmatrix}