Problem 283 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If every element of a third order determinant of value \(\Delta\) is multiplied by 5, then find the value of new determinant.

Step-by-Step Solution

Answer

The value of the new determinant, after each of its elements has been multiplied by 5, is \( \Delta * 5^3 \).

Step 1: Understand the property of determinants

Recognize the property of determinants that states that if each element of a row (or a column) is multiplied by a constant, then the value of determinant gets multiplied by the same constant.

Step 2: Apply determinant property to a third-order determinant

Understand that since this is a third-order determinant (3x3 matrix), the constant being multiplied with every element is 5, and expanded over three rows or columns, it would be \(5^3\).

Step 3: Multiply the initial determinant’s value by the factor

Multiply the given determinant value \(\Delta\) by \(5^3\) to find the final value of the determinant when all of its elements have been multiplied by 5: \(\Delta_{new} = \Delta * 5^3\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying two matrices to produce a new matrix. To multiply matrices, follow these steps:

• Check that the number of columns in the first matrix matches the number of rows in the second matrix. This is crucial for the multiplication to be valid.
• Multiply each element of the rows of the first matrix by the corresponding elements of the columns of the second matrix.
• Add up the products to form an element in the resulting matrix.

This process results in a new matrix where each element represents the sum of products from the row of the first matrix and the column of the second matrix.
Matrix multiplication is not commutative, meaning that the order of matrices in multiplication matters. For example, if matrices A and B are multiplied, the result of AB may not be the same as BA.

Properties of Determinants

Determinants are scalar values that are computed from a square matrix and have several important properties. Understanding these properties is crucial when working with determinants, especially when they are part of solving systems of equations or finding inverses of matrices.

• If you multiply all the elements of a single row or column by a constant, the determinant is also multiplied by that constant.
• If two rows or columns in a determinant are identical, the determinant value is zero. This indicates linear dependence.
• Swapping two rows or two columns of a determinant changes its sign.

These properties can simplify the calculation of determinants by allowing strategic row and column operations. This is especially useful in high-dimensional matrices, where direct computation could be complex.

Third Order Determinants

A third order determinant refers to the determinant of a 3x3 matrix. Calculating such a determinant involves several steps, which make use of the basic properties of determinants.
Let's consider a 3x3 matrix represented as follows:\[\begin{vmatrix}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33} \end{vmatrix}\]To find the determinant of this matrix, use the formula:\[\text{Det } = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]This equation demonstrates how each element of the first row is multiplied by the determinant of the 2x2 submatrix that is not in the element's row or column and are summed up considering alternating signs.
Understanding how third-order determinants are expanded helps in simplifying complex problems, such as transforming matrices or solving linear equations.