CBSE Class 10 Mathematics Matrices and Determinants Notes Set 03

Matrices

A matrix is a rectangular arrangement of numbers in m rows and n columns, called an m × n matrix. The numbers are its elements. A matrix is only an arrangement; it has no numerical value.
1. Square matrix: rows = columns. Its diagonal elements a₁₁, a₂₂, ... form the principal diagonal.
2. Diagonal matrix: all elements off the principal diagonal are zero. Scalar matrix: a diagonal matrix with equal diagonal elements. Identity matrix I: diagonal elements all 1.
3. Zero matrix: all elements 0. Row matrix: one row. Column matrix: one column.

Operations on Matrices

1. Addition and subtraction are done element by element and are defined only for matrices of the same order. Example: \( \begin{bmatrix}5&-1\\3&2\end{bmatrix} + \begin{bmatrix}-4&3\\1&-2\end{bmatrix} = \begin{bmatrix}1&2\\4&0\end{bmatrix} \).
2. Scalar multiplication kA multiplies every element by k: \( 3\begin{bmatrix}5&-1\\3&2\end{bmatrix} = \begin{bmatrix}15&-3\\9&6\end{bmatrix} \).
3. Multiplication AB is possible only if the number of columns of A equals the number of rows of B. The element in row i and column j of AB is the sum of products of row i of A with column j of B. In general AB ≠ BA.
4. Transpose Aᵀ: rows become columns. Trace: the sum of the diagonal elements of a square matrix. Example: for diagonal 3, 0, 9, −5 the trace is 7.

Determinants

A determinant is a number associated with a square arrangement. For second order: \( \begin{vmatrix}a&b\\c&d\end{vmatrix} = ad - bc \). For third order expand along any row or column using the signs + − + / − + − / + − +: \( \begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix} = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2) \).
1. Choose the row or column with most zeros to shorten the work.
2. Multiplying one row or column by k multiplies the determinant by k.
3. Only square matrices have determinants.
Example: \( \begin{vmatrix}2&3&4\\1&0&0\\5&-6&7\end{vmatrix} = -1(21 + 24) = -45 \).

Matrix and Determinant Compared

1. A matrix can have any number of rows and columns; a determinant has equally many rows and columns.
2. A matrix is an arrangement with no value; a determinant has a numerical value.
3. Matrix brackets are [ ] or ( ); determinant bars are | |.