If x + a + b + c = 0, then what is the value of \(\left| {\begin{array}{*{20}{c}} {x + a}&b&c\\ a&{x + b}&c\\ a&b&{x + c} \end{array}} \right|?\)
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Evaluating the Determinant Using Properties
We are asked to find the value of the determinant \(\left| {\begin{array}{*{20}{c}} {x + a}&b&c\\ a&{x + b}&c\\ a&b&{x + c} \end{array}} \right|\) given the condition \(x + a + b + c = 0\).
Let the given determinant be denoted by \(\Delta\):
\(\Delta = \left| {\begin{array}{*{20}{c}} {x + a}&b&c\\ a&{x + b}&c\\ a&b&{x + c} \end{array}} \right|\)
We can use properties of determinants to simplify the calculation. A useful property is that the value of a determinant remains unchanged if we apply column operations of the form \(C_i \leftarrow C_i + kC_j\) or row operations of the form \(R_i \leftarrow R_i + kR_j\).
Let's apply the column operation \(C_1 \leftarrow C_1 + C_2 + C_3\). This means we replace the first column with the sum of the first, second, and third columns. The determinant becomes:
\(\Delta = \left| {\begin{array}{*{20}{c}} {(x + a) + b + c}&b&c\\ {a + (x + b) + c}&{x + b}&c\\ {a + b + (x + c)}&b&{x + c} \end{array}} \right|\)
Simplifying the entries in the first column, we get:
- First row, first column: \(x + a + b + c\)
- Second row, first column: \(x + a + b + c\)
- Third row, first column: \(x + a + b + c\)
So the determinant is now:
\(\Delta = \left| {\begin{array}{*{20}{c}} {x + a + b + c}&b&c\\ {x + a + b + c}&{x + b}&c\\ {x + a + b + c}&b&{x + c} \end{array}} \right|\)
We are given the condition \(x + a + b + c = 0\). Substituting this condition into the determinant, the first column becomes all zeros:
\(\Delta = \left| {\begin{array}{*{20}{c}} {0}&b&c\\ {0}&{x + b}&c\\ {0}&b&{x + c} \end{array}} \right|\)
Another property of determinants states that if any column (or row) of a matrix consists entirely of zeros, then the value of the determinant is zero.
In this case, the first column consists entirely of zeros. Therefore, the value of the determinant is 0.
Thus, \(\left| {\begin{array}{*{20}{c}} {x + a}&b&c\\ a&{x + b}&c\\ a&b&{x + c} \end{array}} \right| = 0\) when \(x + a + b + c = 0\).
Step-by-Step Evaluation Summary
- Start with the given determinant \(\Delta\).
- Apply the column operation \(C_1 \leftarrow C_1 + C_2 + C_3\) to simplify the first column.
- Use the given condition \(x + a + b + c = 0\) to make the first column zero.
- Recognize that a determinant with a column of zeros has a value of zero.
| Original Determinant | Operation | Determinant After Operation | Using Condition | Final Value |
|---|---|---|---|---|
| \(\left| {\begin{array}{*{20}{c}} {x + a}&b&c\\ a&{x + b}&c\\ a&b&{x + c} \end{array}} \right|\) | \(C_1 \leftarrow C_1 + C_2 + C_3\) | \(\left| {\begin{array}{*{20}{c}} {x + a + b + c}&b&c\\ {x + a + b + c}&{x + b}&c\\ {x + a + b + c}&b&{x + c} \end{array}} \right|\) | \(x + a + b + c = 0\) | \(\left| {\begin{array}{*{20}{c}} {0}&b&c\\ {0}&{x + b}&c\\ {0}&b&{x + c} \end{array}} \right| = 0\) |
Revision Table: Key Determinant Properties
| Property | Description | Example |
|---|---|---|
| Row/Column Operations | The value of a determinant remains unchanged if we apply operations \(R_i \leftarrow R_i + kR_j\) or \(C_i \leftarrow C_i + kC_j\). | \(\left| {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right| = \left| {\begin{array}{*{20}{c}} a+kc&b+kd\\ c&d \end{array}} \right|\) (\(R_1 \leftarrow R_1 + kR_2\)) |
| Zero Row/Column | If a row or a column consists entirely of zeros, the determinant is 0. | \(\left| {\begin{array}{*{20}{c}} 0&b\\ 0&d \end{array}} \right| = 0\) |
| Identical Rows/Columns | If two rows or two columns are identical, the determinant is 0. | \(\left| {\begin{array}{*{20}{c}} a&b\\ a&b \end{array}} \right| = 0\) |
Additional Information: Determinant Calculation Basics
A determinant is a scalar value that can be computed from the elements of a square matrix. Determinants are used in various areas of mathematics, including solving systems of linear equations (using Cramer's rule), finding the inverse of a matrix, and calculating areas or volumes.
For a 3x3 matrix \(\left| {\begin{array}{*{20}{c}} a&b&c\\ d&e&f\\ g&h&i \end{array}} \right|\), the determinant can be calculated by expanding along the first row:
\(a \left| {\begin{array}{*{20}{c}} e&f\\ h&i \end{array}} \right| - b \left| {\begin{array}{*{20}{c}} d&f\\ g&i \end{array}} \right| + c \left| {\begin{array}{*{20}{c}} d&e\\ g&h \end{array}} \right|\)
where \(\left| {\begin{array}{*{20}{c}} e&f\\ h&i \end{array}} \right| = ei - fh\), and similarly for the other 2x2 determinants.
While direct expansion is always possible, using properties can significantly simplify the calculation, especially for larger matrices or those with specific structures, as seen in the problem above where the condition \(x + a + b + c = 0\) led to a zero column.
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