Class XII Physics: Matrix Operations

The transpose of a matrix A is obtained by interchanging its rows and columns. It is denoted by A'.

The transpose of the product of two matrices AB is equal to the product of their transposes in reverse order: (AB)' = B'A'.

Taking the transpose twice returns the original matrix: (A')' = A.

The transpose of the sum of two matrices A and B is equal to the sum of their transposes: (A+B)' = A' + B'.

If a matrix A has m rows and n columns, its order is m x n. It is said to be a square matrix if m = n.

A matrix with only one column is called a column matrix.

A matrix with only one row is called a row matrix.

If a matrix has the same number of rows and columns, it is a square matrix.

The sum of two matrices A and B (of the same order) is obtained by adding corresponding elements of A and B.

The product of two matrices A and B is only defined if the number of columns in A is equal to the number of rows in B.

The product of two matrices A and B involves multiplying the corresponding elements in rows of A and columns of B, and then summing the results.

Scalar multiplication of a matrix A by a scalar k is obtained by multiplying each element of A by k.

The identity matrix (I) is a square matrix where the diagonal elements are 1 and all other elements are 0.

A matrix is called null or zero matrix if all its elements are 0.

The determinant of a square matrix is a scalar value that represents certain properties of the matrix.

A matrix is called singular if its determinant is zero.

A matrix is called non-singular if its determinant is non-zero.

The inverse of a square matrix A, denoted as A⁻¹, exists if and only if A is non-singular and satisfies the equation AA⁻¹ = A⁻¹A = I.

The inverse of a 2x2 matrix can often be calculated using a specific formula involving the determinant and the elements of the original matrix.

The inverse of a matrix can be found using methods like Gaussian elimination or row reduction, which involve manipulating the matrix's rows to transform it into the identity matrix.

A matrix is called orthogonal if its transpose is equal to its inverse.

A matrix is called skew-symmetric if its transpose is equal to the negative of the original matrix.

The trace of a square matrix is the sum of its diagonal elements.

Matrix algebra involves operations like addition, subtraction, multiplication, and scalar multiplication. It follows specific rules that might differ from regular algebra.

The order of a matrix refers to its dimensions, expressed as the number of rows by the number of columns.

To find the value of x in the absolute value equation |x| = -10, consider the definition of absolute value. The absolute value of a number is always non-negative.