Under which of the following conditions does the determinant \(\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}\) vanish? 1. a + b + c = 0 2. a 3+ b 3+ c 3= 3abc 3. a 2+ b 2+ c 2- ab - bc - ca = 0 Select the correct answer using the code given below:
1, 2 and 3
Understanding the Determinant and its Vanishing Conditions
The question asks for the conditions under which the determinant of a specific 3x3 matrix becomes zero (vanishes). The given matrix is:
\[ A = \begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix} \]To find when this determinant vanishes, we first need to calculate the determinant of the matrix.
Calculating the Determinant of the Matrix
The determinant of a 3x3 matrix \( \begin{vmatrix}p & q & r \\ s & t & u \\ v & w & x\end{vmatrix} \) is calculated as \( p(tx - uw) - q(sx - uv) + r(sw - tv) \). Applying this formula to our given matrix:
\[ \begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix} = a(c \cdot b - a \cdot a) - b(b \cdot b - a \cdot c) + c(b \cdot a - c \cdot c) \] \[ = a(bc - a^2) - b(b^2 - ac) + c(ab - c^2) \] \[ = abc - a^3 - b^3 + abc + abc - c^3 \] \[ = 3abc - a^3 - b^3 - c^3 \]So, the determinant of the given matrix is \( -(a^3 + b^3 + c^3 - 3abc) \). The determinant vanishes when this expression is equal to zero, which means:
\[ -(a^3 + b^3 + c^3 - 3abc) = 0 \] \[ a^3 + b^3 + c^3 - 3abc = 0 \]This is the core condition for the determinant to vanish. Now, let's examine the given conditions to see which ones satisfy \(a^3 + b^3 + c^3 - 3abc = 0\).
Analyzing Given Conditions for Determinant to Vanish
We are given three conditions:
- \(a + b + c = 0\)
- \(a^3 + b^3 + c^3 = 3abc\)
- \(a^2 + b^2 + c^2 - ab - bc - ca = 0\)
Condition 1: \(a + b + c = 0\)
Consider the algebraic identity: \( a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) \). If \(a + b + c = 0\), then substituting this into the identity:
\[ a^3 + b^3 + c^3 - 3abc = (0)(a^2 + b^2 + c^2 - ab - bc - ca) \] \[ a^3 + b^3 + c^3 - 3abc = 0 \]This shows that if \(a + b + c = 0\), then \(a^3 + b^3 + c^3 - 3abc = 0\). Since the determinant is \( -(a^3 + b^3 + c^3 - 3abc) \), the determinant vanishes if condition 1 is met.
Condition 2: \(a^3 + b^3 + c^3 = 3abc\)
This condition can be rewritten as \( a^3 + b^3 + c^3 - 3abc = 0 \). As shown in the determinant calculation section, this is exactly the condition for the determinant \( -(a^3 + b^3 + c^3 - 3abc) \) to be zero. Therefore, the determinant vanishes if condition 2 is met.
Condition 3: \(a^2 + b^2 + c^2 - ab - bc - ca = 0\)
Again, using the identity: \( a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) \). If \(a^2 + b^2 + c^2 - ab - bc - ca = 0\), then substituting this into the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a+b+c)(0) \] \[ a^3 + b^3 + c^3 - 3abc = 0 \]This shows that if \(a^2 + b^2 + c^2 - ab - bc - ca = 0\), then \(a^3 + b^3 + c^3 - 3abc = 0\). Consequently, the determinant \( -(a^3 + b^3 + c^3 - 3abc) \) vanishes if condition 3 is met.
Furthermore, the expression \(a^2 + b^2 + c^2 - ab - bc - ca\) can be rewritten as \( \frac{1}{2}[(a-b)^2 + (b-c)^2 + (c-a)^2] \). For this expression to be zero, we must have \((a-b)^2 = 0\), \((b-c)^2 = 0\), and \((c-a)^2 = 0\), which implies \(a=b\), \(b=c\), and \(c=a\). Thus, condition 3 is equivalent to \(a=b=c\). If \(a=b=c\), the original determinant becomes:
\[ \begin{vmatrix}a & a & a \\ a & a & a \\ a & a & a\end{vmatrix} \]A matrix with identical rows (or columns) has a determinant of zero. This confirms that if condition 3 is met (i.e., \(a=b=c\)), the determinant vanishes.
Conclusion on Vanishing Determinant Conditions
Based on the analysis:
- Condition 1 (\(a+b+c=0\)) causes the determinant to vanish.
- Condition 2 (\(a^3+b^3+c^3=3abc\)) causes the determinant to vanish.
- Condition 3 (\(a^2+b^2+c^2-ab-bc-ca=0\)) causes the determinant to vanish.
All three conditions lead to the determinant of the given matrix being zero.
Selecting the Correct Answer
We found that the determinant vanishes under condition 1, condition 2, and condition 3. Therefore, the correct answer should include all three conditions.
Looking at the options:
- 1. 1 and 2 only
- 2. 2 and 3 only
- 3. 1 and 3 only
- 4. 1, 2 and 3
Option 4 states that the determinant vanishes under conditions 1, 2, and 3, which aligns with our analysis.
| Condition | Mathematical Statement | Does it cause determinant to vanish? | Reason |
|---|---|---|---|
| 1 | \(a + b + c = 0\) | Yes | Implies \(a^3+b^3+c^3-3abc=0\) |
| 2 | \(a^3 + b^3 + c^3 = 3abc\) | Yes | Equivalent to \(a^3+b^3+c^3-3abc=0\) |
| 3 | \(a^2 + b^2 + c^2 - ab - bc - ca = 0\) | Yes | Implies \(a=b=c\), which implies \(a^3+b^3+c^3-3abc=0\) and determinant is zero. |
Revision Table: Determinant Vanishing Conditions
Here's a quick summary of the key points related to the determinant vanishing:
- The determinant is \( -(a^3 + b^3 + c^3 - 3abc) \).
- The determinant is zero if and only if \( a^3 + b^3 + c^3 - 3abc = 0 \).
- The identity \( a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) \) is crucial.
- If \(a+b+c=0\), the determinant vanishes.
- If \(a^3+b^3+c^3=3abc\), the determinant vanishes.
- If \(a^2+b^2+c^2-ab-bc-ca=0\) (which implies \(a=b=c\)), the determinant vanishes.
Additional Information: Properties of Determinants and Algebraic Identities
Determinants have several useful properties. For a 3x3 matrix, the determinant being zero indicates that the matrix is singular, meaning its columns (or rows) are linearly dependent. In simpler terms, one column (or row) can be expressed as a combination of the others.
The algebraic identity used, \( a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) \), is a fundamental identity in algebra. It's often used in problems involving symmetric expressions in three variables. The term \(a^2 + b^2 + c^2 - ab - bc - ca\) is also related to the sum of squares of differences: \( a^2 + b^2 + c^2 - ab - bc - ca = \frac{1}{2}[(a-b)^2 + (b-c)^2 + (c-a)^2] \). This form clearly shows that \(a^2 + b^2 + c^2 - ab - bc - ca = 0\) if and only if \(a=b=c\).
Understanding these identities and determinant properties is key to solving problems like this one involving determinant vanishing conditions.
Important Questions from Determinants
If \(A=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 3 & 0 \\ 1 & 0 & 1 \end{array}\right]\), then what is the value of det[adj(adjA)] ?
If A, B and C are square matrices of order 3 and det(BC) = 2 det(A), then what is the value of det(2A-1BC)?
If \(A=\left[\begin{array}{rrr} 0 & 3 & 4 \\ -3 & 0 & 5 \\ -4 & -5 & 0 \end{array}\right]\), then which one of the following statements is correct?
If \(\left|\begin{array}{ccc} x^2+3 x & x-1 & x+3 \\ x+1 & -2 x & x-4 \\ x-3 & x+4 & 3 x \end{array}\right|\) = ax4 + bx3 + cx2 + dx + e, then what is the value of e?"
If all elements of a third order determinant are equal to 1 or -1, then the value of the determinant is:
