Matrices Class 12 Notes- Chapter 3

What are Matrices?

Matrices are an important branch of mathematics, which also helps students understand other mathematical concepts due to the interrelatedness to other branches. A matrix can be defined as an array of numbers or functions arranged in a rectangular order.

A matrix which possesses rows titled “X” and columns titled as “Y” is called as a matrix of an order of X x Y. The major operations carried out with Matrices are the Addition, Subtraction, and Multiplication. Understanding the types of matrices is essential to understand, the other topics better.

What are the types of Matrices?

Matrices are commonly classified into:

    • Column Matrix:- A matrix which consists of a singular column can be defined as a column matrix.
    • Row Matrix:- If a matrix has only a singular row, then it is termed as a row matrix
    • Square Matrix:- When the number of rows is equal to the number of columns, then the matrix is defined as a square matrix.
    • Diagonal Matrix:- If the diagonal elements are zero in a square matrix, then it can be defined as a diagonal matrix.
    • Scalar Matrix:- When the diagonal elements are equal in a diagonal matrix, then this matrix is assumed to be a Scalar matrix.
    • Identity Matrix:- An identity matrix can be defined as a square matrix, whose elements in the diagonal are 1, while the other elements are zero.
    • Zero Matrix:- In a matrix, when all the elements are zero, it can simply be defined as a null matrix or a zero matrix.

If two square matrices called A and B exist then we can safely assume that AB = BA = 1, where B is the inverse matrix of A and A is the inverse of B. A square matrix, on the other hand, can be termed as the sum of a skew matrix and a symmetric matrix.

What is determinant of matrix?

A determinant of a matrix is the addition of the products of the elements within a square matrix. If p, q, r, s are the elements in a matrix.

\(\left | A \right |\) = \(\begin{bmatrix} p & q\\ s & r\end{bmatrix}\)

Then, the determinant is (ps-qr)

What is inverse of matrix?

The inverse function is obtained by simply swapping the position of terms. If we take the same terms of the above matrix then, A= \(\begin{bmatrix} p & q\\ s & r\end{bmatrix}\) has the inverse as \(A^{-1}=1/\left | A \right |\begin{bmatrix} s & -q\\ -r & p \end{bmatrix}\)

 

Important Questions

  1. If the matrix P is both symmetric and skew symmetric, then (A) P is a diagonal matrix (B) P is a zero matrix (C) P is a square matrix (D) None of these
  2. If unit sale prices of p, q and r are ` 3.50, ` 2.50 and ` 1.00, respectively,
    find the total revenue in each market with the help of matrix algebra.
  3. Find the gross profit. If the unit costs of the above three commodities are ` 3.00, ` 2.00 and 100 paise respectively.
  4. If P and Q are symmetric matrices, prove that PQ-QP is a skew symmetric
    Matrix.
  5. Show that the matrix P= \(\begin{bmatrix} 1 &-1 &5 \\ -1 &2 &1 \\ 5 &1 &3 \end{bmatrix}\)

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