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Chapter Analysis
Intermediate42 pages • EnglishQuick Summary
This chapter introduces matrices as a powerful mathematical tool used to solve systems of linear equations and perform various operations such as magnification, rotation, and reflection. It describes matrices as ordered arrays of numbers or functions, emphasizing their importance in fields like electronic spreadsheets, business, science, genetics, and economics. The chapter covers the basic operations and properties of matrices, including addition, multiplication, and finding inverses.
Key Topics
- •Order of a matrix
- •Types of matrices
- •Matrix operations
- •Inverse of a matrix
- •Applications of matrices
- •Properties of matrix addition
- •Scalar multiplication of a matrix
- •Elementary row and column operations
Learning Objectives
- ✓Understand the definition and order of matrices
- ✓Identify different types of matrices
- ✓Perform basic matrix operations such as addition and multiplication
- ✓Calculate the inverse of a matrix
- ✓Apply matrices to solve real-world problems
- ✓Understand and prove matrix properties
Questions in Chapter
If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
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Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
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Find the values of x, y, z if the matrix satisfies the equation A′A = I.
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For what values of x does the equation hold for the given matrices?
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If A is given, show that A² – 5A + 7I = 0.
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Additional Practice Questions
What is a matrix and what are its applications?
easyAnswer: A matrix is an ordered rectangular array of numbers, used to represent data or equations. Applications include system of equations, computer graphics, economics, and genetics.
Describe the terms 'order of a matrix' and 'types of matrices'.
mediumAnswer: The order of a matrix is m x n, where m is the number of rows and n the number of columns. Types include row matrices, column matrices, square matrices, diagonal matrices, etc.
Explain how to find the inverse of a matrix.
hardAnswer: To find the inverse of a matrix A, it must be square and its determinant must not be 0. The inverse is found through various methods, including row reduction or using an adjugate matrix.
How do you perform matrix addition and multiplication?
mediumAnswer: Matrix addition, A + B, requires matrices of the same size, adding corresponding elements. Matrix multiplication, AB, occurs when the number of columns in A equals the number of rows in B, combining row elements with column elements.
What is the significance of the identity matrix in matrix operations?
easyAnswer: The identity matrix acts as a multiplicative identity in matrices, similar to the number 1 in arithmetic, leaving other matrices unchanged when multiplied.