Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

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1:36 minutes

Problem 1

Textbook Question

Answer each question. What is the product of [2x2 matrix] and I2 (in either order)?

Verified step by step guidance

Identify the given matrix, which is a 2x2 matrix. Let's denote it as \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).

Recognize that \( I_2 \) is the 2x2 identity matrix, which is \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \).

Recall the property of the identity matrix: when any matrix is multiplied by the identity matrix, the result is the original matrix.

Multiply the given matrix \( A \) by the identity matrix \( I_2 \) using matrix multiplication rules: \( A \times I_2 = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \).

Verify that the product of \( A \) and \( I_2 \) is \( A \) itself, confirming the identity property: \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix. The product of a matrix A (of size m x n) and a matrix B (of size n x p) results in a matrix C (of size m x p). Each element of the resulting matrix is calculated as the dot product of the corresponding row from the first matrix and the column from the second matrix.

Identity Matrix

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. For example, the 2x2 identity matrix I2 is represented as [[1, 0], [0, 1]]. When any matrix is multiplied by the identity matrix of compatible size, the original matrix remains unchanged, making the identity matrix a fundamental element in linear algebra.

Order of Multiplication

In matrix multiplication, the order in which matrices are multiplied matters. Generally, the product of matrices A and B (AB) is not the same as the product of B and A (BA), unless one of the matrices is an identity matrix. Understanding this concept is crucial for correctly performing operations involving matrices and ensuring accurate results.