Understanding Determinants and Laplace Expansion

Let's analyze the given Assertion and Reason regarding determinants of matrices.

Analyzing Assertion (A)

Assertion (A) states that for the matrix \(A = \left(\begin{array}{ccc}5 & 0 & 3 \\ -1 & 0 & 2 \\ 1 & 0 & 1\end{array}\right)\), the determinant det A = 0 because all elements in column II are zero.

To verify this, we can use the property of determinants which states: If a matrix has a column (or a row) consisting entirely of zero elements, then its determinant is zero.

Looking at matrix A:

We can clearly see that all the elements in the second column (Column II) are 0 (i.e., 0, 0, 0). Based on the determinant property, the determinant of matrix A should be 0. Therefore, Assertion (A) is correct.

Analyzing Reason (R)

Reason (R) states that Laplace expansion permits evaluation of a determinant along any row or column.

Laplace expansion, also known as cofactor expansion, is a method for calculating the determinant of a square matrix. It involves summing the products of the elements of any row or column with their corresponding cofactors. The great advantage of this method is that you can choose *any* row or *any* column to perform the expansion. This is particularly useful when a matrix has a row or column with many zero elements, as it simplifies the calculation.

For a \(3 \times 3\) matrix like A, expanding along column II (the zero column) using Laplace expansion would give:

det A = \(a_{12}C_{12} + a_{22}C_{22} + a_{32}C_{32}\)

Where \(a_{ij}\) are the elements and \(C_{ij}\) are their cofactors.

Since \(a_{12}=0\), \(a_{22}=0\), and \(a_{32}=0\), the expansion becomes:

det A = \(0 \times C_{12} + 0 \times C_{22} + 0 \times C_{32}\) = \(0 + 0 + 0\) = 0

This calculation confirms the property stated in Assertion (A).

Reason (R) correctly describes Laplace expansion and its flexibility. Thus, Reason (R) is correct.

Relationship between Assertion (A) and Reason (R)

We have established that both Assertion (A) and Reason (R) are correct statements.

Assertion (A) states that the determinant is zero because of a zero column. Reason (R) describes Laplace expansion, a method to calculate determinants by expanding along any row or column.

The property mentioned in Assertion (A) (determinant being zero due to a zero column) can be directly explained and proven using the method described in Reason (R) (Laplace expansion along that specific zero column). Expanding along the zero column using Laplace expansion results in every term being zero, thus proving the determinant is zero.

Therefore, Reason (R) provides a correct explanation for Assertion (A).

Conclusion

Both the Assertion (A) and the Reason (R) are correct statements, and Reason (R) correctly explains why Assertion (A) is true.

Revision Table: Determinant Properties

Additional Information: Laplace Expansion and Cofactors

Laplace expansion is a fundamental method for calculating determinants, especially useful for smaller matrices or matrices with many zeros.

• Minor: The minor \(M_{ij}\) of an element \(a_{ij}\) in a square matrix is the determinant of the submatrix formed by deleting the i-th row and j-th column.
• Cofactor: The cofactor \(C_{ij}\) of an element \(a_{ij}\) is given by \(C_{ij} = (-1)^{i+j} M_{ij}\). The term \((-1)^{i+j}\) determines the sign pattern of the cofactors (often visualized as a checkerboard pattern of + and -).
• Expansion along a Row: The determinant of an \(n \times n\) matrix A along row i is given by:
  det A = \(\sum_{j=1}^{n} a_{ij} C_{ij} = a_{i1}C_{i1} + a_{i2}C_{i2} + \dots + a_{in}C_{in}\).
• Expansion along a Column: The determinant of an \(n \times n\) matrix A along column j is given by:
  det A = \(\sum_{i=1}^{n} a_{ij} C_{ij} = a_{1j}C_{1j} + a_{2j}C_{2j} + \dots + a_{nj}C_{nj}\).
• Choosing a row or column with the most zeros simplifies the calculation significantly because terms where \(a_{ij} = 0\) become zero, and their cofactors do not need to be calculated.