NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.4

To correctly multiply two matrices, A and B, as per the NCERT methodology, you must first ensure they are compatible for multiplication: the number of columns in matrix A must equal the number of rows in matrix B. If this condition isn't met, the product is not defined.

The steps are as follows:

• To find the element in the i-th row and j-th column of the product matrix, select the i-th row of matrix A and the j-th column of matrix B.
• Multiply their corresponding elements one by one.
• Sum up all these products. This final sum gives you the single element for that specific position in the resulting matrix.

To find the inverse of a square matrix A using elementary row operations, start by writing the equation A = IA, where I is the identity matrix of the same order as A.

The step-by-step method is as follows:

• Apply a sequence of elementary row operations to the matrix A on the left-hand side (LHS) of the equation to transform it into the identity matrix I.
• Simultaneously, apply the exact same sequence of operations to the identity matrix I on the right-hand side (RHS).
• Once the matrix A on the LHS becomes the identity matrix I, the matrix on the RHS will be transformed into the inverse of A, or A⁻¹.

The final equation will be I = A⁻¹A. If you obtain a row or column of all zeros on the LHS at any stage, then A⁻¹ does not exist.

An invertible matrix is a square matrix 'A' for which another square matrix 'B' of the same order exists, such that their product is the identity matrix (I). This is expressed as AB = BA = I. The matrix B is called the inverse of A, written as A⁻¹.

According to the NCERT syllabus, you can determine if a matrix is invertible in two main ways:

• Using elementary operations: A square matrix A is invertible if it can be reduced to the identity matrix (I) by applying a sequence of elementary operations.
• Using determinants (Chapter 4): A square matrix A is invertible if and only if its determinant is non-zero (det(A) ≠ 0).

The NCERT solutions for Chapter 3 are based on three specified elementary row operations (or transformations) used to find a matrix's inverse. They are:

• Interchange of any two rows: Swapping the positions of any two rows (denoted as Rᵢ ↔ Rⱼ).
• Multiplication of a row by a non-zero scalar: Multiplying all elements of a single row by any non-zero constant 'k' (denoted as Rᵢ → kRᵢ).
• Addition of a multiple of one row to another: Adding the elements of one row, multiplied by a non-zero scalar 'k', to the corresponding elements of another row (denoted as Rᵢ → Rᵢ + kRⱼ).

These same three operations can also be applied to columns.

Matrix multiplication is not always commutative due to its row-by-column definition. Unlike the multiplication of real numbers, the order in which matrices are multiplied significantly changes the result. There are two primary reasons:

• Existence of the Product: The product AB might be defined while BA is not. For instance, if A is a 2x3 matrix and B is a 3x4 matrix, AB is a valid 2x4 matrix. However, BA is undefined as the number of columns in B (4) does not match the number of rows in A (2).
• Different Resulting Matrices: Even when both AB and BA are defined (e.g., for two square matrices of the same order), the resulting matrices are generally not equal because the calculation process yields different elements when the order is reversed. This property is known as non-commutativity.

A rectangular matrix cannot have an inverse because the definition of an inverse requires that for a matrix A, there exists a matrix B such that AB = BA = I, where I is the identity matrix. This condition can only be satisfied if both A and B are square matrices of the same order.

If A is a rectangular matrix of order m x n (where m ≠ n), any potential inverse B would have to be of order n x m. The product AB would be an m x m matrix, while the product BA would be an n x n matrix. Since m ≠ n, the resulting matrices AB and BA have different orders and can never be equal, making an inverse impossible.

The inverse matrix provides a direct method for solving a system of linear equations. A system can be represented in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

If the coefficient matrix A is invertible and its inverse A⁻¹ exists, you can find the solution by pre-multiplying both sides of the equation by A⁻¹:

A⁻¹(AX) = A⁻¹B

Since A⁻¹A = I (the identity matrix), the equation simplifies to IX = A⁻¹B, which gives the unique solution for the variables as X = A⁻¹B.

The transpose of a matrix (A' or Aᵀ) is an operation where the matrix's rows and columns are interchanged. Its significance lies in its use for defining special types of matrices as per the NCERT syllabus.

The transpose helps define:

• Symmetric Matrix: A square matrix A is symmetric if it is equal to its own transpose (A' = A). This means the element at position (i, j) is equal to the element at (j, i).
• Skew-Symmetric Matrix: A square matrix A is skew-symmetric if it is the negative of its transpose (A' = -A). This means the element at (i, j) is the negative of the element at (j, i), and all diagonal elements must be zero.

A key theorem in Chapter 3 also states that any square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.