Calculating the Determinant and Finding k

The question asks us to find the value of \(k\) given a determinant equation. We are given the equation \(\rm \begin{vmatrix} a & -b & a - b - c\\ -a & b & -a + b - c\\ -a & -b & -a - b + c \end{vmatrix} - kabc = 0\), where \(a \ne 0, b \ne 0, c \ne 0\). To find \(k\), we first need to evaluate the given determinant.

Evaluating the Determinant

Let the determinant be denoted by \(D\). The matrix is:

We can simplify the determinant calculation by using row operations. Applying elementary row operations does not change the value of the determinant.

• Operation 1: Add Row 1 to Row 2 (\(R_2 \leftarrow R_2 + R_1\))
• Operation 2: Add Row 1 to Row 3 (\(R_3 \leftarrow R_3 + R_1\))

Let's perform these operations:

New Row 2:

• Column 1: \(-a + a = 0\)
• Column 2: \(b + (-b) = 0\)
• Column 3: \((-a + b - c) + (a - b - c) = -a + b - c + a - b - c = -2c\)

New Row 3:

• Column 1: \(-a + a = 0\)
• Column 2: \(-b + (-b) = -2b\)
• Column 3: \((-a - b + c) + (a - b - c) = -a - b + c + a - b - c = -2b\)

The determinant becomes:

Now we can easily expand the determinant along the first column (Column 1), as it has two zero entries. The determinant is given by:

\(D = a \times \begin{vmatrix} 0 & -2c \\ -2b & -2b \end{vmatrix} - 0 \times (\text{minor}) + 0 \times (\text{minor})\)

Now we calculate the 2x2 determinant:

\(\begin{vmatrix} 0 & -2c \\ -2b & -2b \end{vmatrix} = (0 \times (-2b)) - ((-2c) \times (-2b)) = 0 - (4bc) = -4bc\)

So, the value of the determinant \(D\) is:

\(D = a \times (-4bc) = -4abc\)

Solving for k

The given equation is \(\rm \begin{vmatrix} a & -b & a - b - c\\ -a & b & -a + b - c\\ -a & -b & -a - b + c \end{vmatrix} - kabc = 0\).

Substitute the calculated value of the determinant, \(D = -4abc\), into the equation:

\(-4abc - kabc = 0\)

We are given that \(a \ne 0, b \ne 0, c \ne 0\). This means \(abc \ne 0\). We can divide the entire equation by \(abc\):

\(-4 - k = 0\)

Now, solve for \(k\):

\(-k = 4\)

\(k = -4\)

Thus, the value of \(k\) is -4.

Comparing with Options

Let's compare our result with the given options:

• Option 1: -4
• Option 2: -2
• Option 3: 2
• Option 4: 4

Our calculated value \(k = -4\) matches Option 1.

Revision Table: Determinant Calculation Key Points

Additional Information: Properties of Determinants

Determinants have several useful properties that can simplify calculations:

• The determinant of a matrix is equal to the determinant of its transpose (\(\det(A) = \det(A^T)\)).
• If any row or column consists entirely of zeros, the determinant is zero.
• If two rows or two columns are identical or proportional, the determinant is zero.
• If a row or column is multiplied by a scalar \(c\), the determinant is multiplied by \(c\).
• If two rows or columns are swapped, the determinant changes sign.
• The determinant of a product of matrices is the product of their determinants (\(\det(AB) = \det(A) \det(B)\)).
• The determinant of an upper or lower triangular matrix is the product of its diagonal elements. The simplified matrix we obtained after row operations was an upper triangular matrix after swapping R2 and R3 (or considering the effective structure for expansion).