Consider the determinant Δ = \(\left|\begin{array}{lll}\text{a}_{11} & \text{a}_{12} & \text{a}_{13} \\ \text{a}_{21} & \text{a}_{22} & \text{a}_{23} \\ \text{a}_{31} & \text{a}_{32} & \text{a}_{33}\end{array}\right|\) If a 13 = yz, a 23 = zx, a 33 = xy and the minors of a 13 , a 23 , a 33 are respectively (z − y), (z − x), (y − x) then what is the value of Δ ?
(z − y) (z − x) (y − x)
Calculating the Value of a Determinant Using Minors and Expansion
The problem asks us to find the value of a 3x3 determinant \(\Delta\) given specific values for the elements in the third column (\(a_{13}, a_{23}, a_{33}\)) and their corresponding minors (\(M_{13}, M_{23}, M_{33}\)).
The determinant is given as:
\[ \Delta = \left|\begin{array}{lll} \text{a}_{11} & \text{a}_{12} & \text{a}_{13} \\ \text{a}_{21} & \text{a}_{22} & \text{a}_{23} \\ \text{a}_{31} & \text{a}_{32} & \text{a}_{33} \end{array}\right| \]We are provided with the following information:
- \(a_{13} = yz\)
- \(a_{23} = zx\)
- \(a_{33} = xy\)
- Minor of \(a_{13}\), \(M_{13} = (z - y)\)
- Minor of \(a_{23}\), \(M_{23} = (z - x)\)
- Minor of \(a_{33}\), \(M_{33} = (y - x)\)
Determinant Expansion Using Cofactors
The value of a determinant can be calculated by expanding along any row or column. Expanding along the third column is convenient because we are given the elements and their minors in this column. The expansion formula for a 3x3 determinant along the third column is:
\[ \Delta = a_{13}C_{13} + a_{23}C_{23} + a_{33}C_{33} \]where \(C_{ij}\) is the cofactor of the element \(a_{ij}\). The cofactor is related to the minor \(M_{ij}\) by the formula \(C_{ij} = (-1)^{i+j}M_{ij}\).
Let's calculate the cofactors for the elements in the third column:
- \(C_{13} = (-1)^{1+3}M_{13} = (-1)^4 M_{13} = +M_{13}\)
- \(C_{23} = (-1)^{2+3}M_{23} = (-1)^5 M_{23} = -M_{23}\)
- \(C_{33} = (-1)^{3+3}M_{33} = (-1)^6 M_{33} = +M_{33}\)
Now, substitute the cofactors back into the expansion formula:
\[ \Delta = a_{13}(+M_{13}) + a_{23}(-M_{23}) + a_{33}(+M_{33}) \] \[ \Delta = a_{13}M_{13} - a_{23}M_{23} + a_{33}M_{33} \]Substituting Given Values to Find Determinant Value
Now, substitute the given values of \(a_{ij}\) and \(M_{ij}\) into the formula:
- \(a_{13} = yz\), \(M_{13} = (z - y)\)
- \(a_{23} = zx\), \(M_{23} = (z - x)\)
- \(a_{33} = xy\), \(M_{33} = (y - x)\)
Let's expand and simplify this expression:
\[ \Delta = (yz \cdot z - yz \cdot y) - (zx \cdot z - zx \cdot x) + (xy \cdot y - xy \cdot x) \] \[ \Delta = (yz^2 - y^2z) - (z^2x - zx^2) + (xy^2 - x^2y) \] \[ \Delta = yz^2 - y^2z - z^2x + zx^2 + xy^2 - x^2y \]Let's rearrange the terms in a cyclic order for easier pattern recognition:
\[ \Delta = xy^2 - x^2y + yz^2 - y^2z + zx^2 - z^2x \]Comparing with Options and Factoring
We need to compare this expanded form with the given options, which are in factored form. Let's expand the first option: \((z - y) (z - x) (y - x)\)
First, multiply the first two factors:
\[ (z - y)(z - x) = z(z - x) - y(z - x) = z^2 - zx - yz + yx \]Now, multiply the result by the third factor \((y - x)\):
\[ (z^2 - zx - yz + yx)(y - x) \] \[ = z^2(y - x) - zx(y - x) - yz(y - x) + yx(y - x) \] \[ = (z^2y - z^2x) - (zxy - zx^2) - (y^2z - yzx) + (y^2x - yx^2) \] \[ = z^2y - z^2x - zxy + zx^2 - y^2z + yzx + y^2x - yx^2 \]Combine terms and rearrange:
\[ = z^2y - z^2x - zxy + zx^2 - y^2z + xyz + xy^2 - x^2y \] \[ = z^2y - z^2x - y^2z + zx^2 + xy^2 - x^2y \]Rearranging terms to match the order of \(\Delta\) we found:
\[ = xy^2 - x^2y + yz^2 - y^2z + zx^2 - z^2x \]This expanded form matches the expanded form of \(\Delta\) that we calculated using the determinant expansion. Therefore, the value of the determinant is \((z - y) (z - x) (y - x)\).
Summary of Calculation Steps
- Identify the given determinant setup, including specific elements and their minors in the third column.
- Recall the formula for determinant expansion along a column, using elements and cofactors.
- Calculate the cofactors from the given minors, remembering the \( (-1)^{i+j} \) factor.
- Substitute the elements and cofactors (or minors with appropriate signs) into the expansion formula.
- Expand and simplify the resulting algebraic expression.
- Compare the simplified expression with the given options. If options are factored, expand them to verify the match.
Revision Table: Determinant Properties
| Concept | Description | Formula/Example |
|---|---|---|
| Determinant | A scalar value calculated from the elements of a square matrix. | For a 2x2 matrix \(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right| = ad - bc\) |
| Minor (\(M_{ij}\)) | The determinant of the submatrix formed by deleting the i-th row and j-th column. | For \(a_{11}\) in 3x3, \(M_{11} = \left|\begin{array}{ll}a_{22} & a_{23} \\ a_{32} & a_{33}\end{array}\right|\) |
| Cofactor (\(C_{ij}\)) | Related to the minor by \(C_{ij} = (-1)^{i+j}M_{ij}\). The sign depends on the position (i+j) being even or odd. | \(C_{11} = +M_{11}\), \(C_{12} = -M_{12}\) |
| Determinant Expansion | Calculating the determinant by summing the products of elements of a row/column with their corresponding cofactors. | \(\Delta = a_{i1}C_{i1} + a_{i2}C_{i2} + \dots\) (along row i) or \(\Delta = a_{1j}C_{1j} + a_{2j}C_{2j} + \dots\) (along column j) |
Additional Information: Cyclic Expressions and Factoring
The expression \(xy^2 - x^2y + yz^2 - y^2z + zx^2 - z^2x\) is a common cyclic expression in algebra. Recognizing such patterns can be helpful. This specific expression factors into products involving differences of the variables. The factors are often of the form \((x-y), (y-z), (z-x)\) or variations with signs swapped.
In this case, the factoring resulted in \((z - y)(z - x)(y - x)\). It's useful to remember the expansion of common cyclic products. For instance, \((x-y)(y-z)(z-x) = xy^2 - x^2y + yz^2 - y^2z + zx^2 - z^2x\).
Being familiar with common algebraic identities and factorization patterns is crucial for solving problems involving symbolic determinants.
Important Questions from Application of Determinants
The system of linear equation kx + y + z = 1, x + ky + z = 1 and x + y + kz = 1 has a unique solution under which one of the following conditions?
Which of the following are correct in respect of the system of equation
x + y + z = 8,
x – y + 2z = 6 and
3x – y + 5z = k?
1. They have no solution if k = 15
2. They have infinitely many solutions, if k = 20
3. They have a unique solution if k = 25
Select the correct answer using the code given below:Under what condition does the above system of equations have unique solutions?
- The number of values of $k$, for which the system of equations: $(k^2 - 4)x + (k - 2)y = k^2 - 2k$ and $(k + 2)x + y = k$ have infinitely many solutions, is -
For what values of k is the system of equations 2k 2x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 consistent?
