Chapter 02: Matrices and Determinants
Notes (Solutions) of Chapter 02: Matrices and Determinants, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), National Foundation Book, NBF, Federal Board (FBISE)
A very common use of matrices in daily life is encryption. We use them to scramble data for security purpose and to encode and decode this data. There is a key that helps encode and decode data which is generated by matrices. The screen of any electronic device, like smart phone or LED TV screen is essentially a pixel matrix. When we rotate the phone and it is in landscape form. The matrix is actually rotated using the transpose. When we touch the screen of a cell phone at some specific position; the position is calculated by matrix properties.
A matrix is an array or number arranged in horizontal and vertical lines enclosed within square brackets, Matrices are usually denoted with capital letters. The horizontal lines are known as rows of the matrix and vertical lines are known as columns of the matrix.
Contents & summary:
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Introduction
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Order of matrix, square of matrix, rectangular matrix, row matrix, column matrix, null matrix, identity matrix, , diagonal matrix, scalar matrix, identity matrix, lower triangular matrix, upper triangular matrix, transpose of a matrix, symmetric or skew symmetric of a matrix
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Addition of matrix , subtraction of matrix, multiplication of matrix, commutative law, associative law,
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Find Determinant of 2*2 matrices, Determinant of 3*3 matrices, Determinant of 3*3 matrices by co-factor method. (a_11 A_11+a_12 A_12+a_13 A_13 and A_ij=(-1)^(i+j) M_ij). Find adjoint of 3*3 matrices, and how to find inverse of 3*3 matrices.
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|A|=|A|^t. If any two rows (or columns) of a square matrix A are interchanged, then the resulting matrix is B then |B|=-|A|. If any two rows (or columns) of a square matrix are identical then the value of determinant is zero. If all the elements of a row or a column of a square matrix, then the value of determinant is zero. If we multiply each element of a row or a column with a non-zero scalar k then the resulting matrix is B and |B|=k|A|
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Echelon Form, Reduced Echelon Form, Rank of matrix, Find Inverse by row operation.
- Exercise 2.6
- Homogeneous Linear equation, Gauss Elimination method, Gauss-Jordan method, Crammer’s rule, Matrix Inversion method, Consistent Inconsistent, Encoded Decoded.
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