Problem 32 - Algebra for JEE ADVANCED

Question

If \(A\) and \(B\) are two non-singular matrices such that \(A B=C\), then \(|B|\) is equal to a. \(\frac{|C|}{|A|}\) b. \(\frac{|A|}{|C|}\) c. \(\mid C\) d. none of these

Step-by-Step Solution

Answer

Option 'a' is the correct answer: \(\frac{|C|}{|A|}\).

Step 1: Understand the Relationship Between Determinants

Since \(A\) and \(B\) are non-singular matrices and given that \(A B = C\), the relationship between their determinants is given by the property \(|A B| = |A| \cdot |B|\). So, \(|C| = |A| \cdot |B|\).

Step 2: Solve for \(|B|\)

We want to find the determinant \(|B|\). From the equation \(|C| = |A| \cdot |B|\), we can solve for \(|B|\) by dividing both sides by \(|A|\). Thus \(|B| = \frac{|C|}{|A|}\).

Step 3: Confirm the Correct Answer

Based on the calculation \(|B| = \frac{|C|}{|A|}\), we compare our result with the provided options. Option 'a' specifies \(\frac{|C|}{|A|}\), which matches our solution. Therefore, option 'a' is the correct choice.

Key Concepts

Non-singular Matrices

In the world of linear algebra, understanding non-singular matrices is crucial. A non-singular matrix, also known as an invertible matrix, is a square matrix with some special properties. One key property is that the determinant of a non-singular matrix is never zero. This is significant because if a matrix has a determinant of zero, it means the matrix cannot be inverted. In other words, you can't "undo" its effects when applied to a vector.
When dealing with non-singular matrices:

• Every non-singular matrix has a unique inverse, which when multiplied by the matrix gives the identity matrix.
• This unique inverse is important for solving linear equations, transforming spaces, and understanding geometric transformations.
• Non-singular matrices represent linear transformations that are bijective, meaning they are both injective (one-to-one) and surjective (onto).

Remember, for any matrix, knowing whether it is singular or non-singular helps you decide the kind of mathematical operations you can perform with it. This is especially useful when considering matrix multiplication and solving systems of linear equations.

Determinant Properties

Determinants are a fundamental concept when discussing matrices, especially in calculating matrix properties like area scaling, checking for invertibility, and more.
The determinant of a matrix is a special number that provides a lot of information about the matrix itself.
Some important properties of determinants include:

• For any square matrices A and B, the determinant of their product is the product of their determinants:
  \(|A imes B| = |A| imes |B|\).
• A matrix with a determinant of zero is singular and cannot be inverted.
• Switching two rows or columns of a matrix results in the determinant changing sign. This property is crucial in understanding how row operations affect matrices.
• If any row or column of a matrix is multiplied by a scalar, the determinant is multiplied by that scalar.

Determinants are not always straightforward to calculate as the size of the matrix increases, but these properties are vital tools for simplifying the calculations you need in solving matrix-related problems.

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra, allowing us to combine linear transformations and solve complex systems of equations.
It's not just about multiplying numbers; it's a series of dot products between rows and columns.
When multiplying matrices:

• The number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be defined.
• The resulting matrix has dimensions equal to the rows of the first matrix and the columns of the second matrix.
• This operation is associative, meaning that the way you group the matrices (e.g., \((A imes B) imes C\) is the same as \(A imes (B imes C)\)) does not change the result.
• However, it is not commutative;
  \(A imes B\) is generally not the same as \(B imes A\).

Matrix multiplication is invaluable when multiple linear transformations are applied. It allows us to concisely express the combination of operations, showing how one transformation followed by another affects a space.