Inverse of a Matrix and Systems of Linear Equations

Name: Inverse of a Matrix and Systems of Linear Equations
Author: StemPy Lab

The adjoint and the inverse of a matrix, and the test for a matrix that has no inverse. Then solving systems of two and three equations with the inverse, deciding when there is one solution, none or infinitely many, and finding the inverse by row operations. Cramer's rule and models from real life are included.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Adjoint and inverse of a matrix
A square matrix $A$ has an inverse $A^{-1}$ when $AA^{-1} = A^{-1}A = I$ . The adjoint of $A$ is the transpose of its cofactor matrix, and $A\,\text{adj}\,A = |A|\,I$ . So $A^{-1} = \dfrac{\text{adj}\,A}{|A|}$ , which exists exactly when $|A| \ne 0$ . Such a matrix is non-singular, and one with $|A| = 0$ is singular. The inverse of a product reverses the order of the factors.
1 simulation1 practice question1 Python program4 problems1 Python lab
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Lesson 2about 25 min
Solving systems with the inverse and Cramer's rule
A system of linear equations can be written $AX = B$ . If $|A| \ne 0$ the system has exactly one solution, $X = A^{-1}B$ . If $|A| = 0$ there is either no solution or infinitely many, and $(\text{adj}\,A)B$ helps to decide. A homogeneous system $AX = O$ has non-zero solutions exactly when $|A| = 0$ . Cramer's rule gives each unknown as a ratio of two determinants.
1 simulation1 practice question1 Python program4 problems
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Lesson 3about 25 min
Row operations and linear models
There are three elementary row operations: swap two rows, multiply a row by a non-zero number, and add a multiple of one row to another. Applying the operations that turn $A$ into $I$ to the matrix $I$ gives $A^{-1}$ , so $[A\,|\,I]$ becomes $[I\,|\,A^{-1}]$ . The same operations on an augmented matrix solve a system. Linear systems also model traffic flows and the output of industries.
1 simulation1 practice question1 Python program2 problems1 Python lab1 question set1 note
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Inverse and Linear Systems: Chapter Test

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Inverse of a Matrix and Systems of Linear Equations

Course outline (3 lessons)

Adjoint and inverse of a matrix

Solving systems with the inverse and Cramer's rule

Row operations and linear models

Test Your Learning

Your teacher

Discussion (0)