Textbook Question
For each pair of matrices A and B, find (a) AB and (b) BA. See Example 7.
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7. Systems of Equations & Matrices
Determinants and Cramer's Rule
6:15 minutesProblem 84
Textbook Question
Find AB and BA for the following matrices.
Matrix B acts as the multiplicative element for 2 2 square matrices.
Verified step by step guidance1
First, identify the matrices A and B explicitly. Since the problem statement does not provide the actual entries of matrices A and B, you need to write down or obtain the elements of both matrices. Each should be a 2x2 matrix, for example, \( A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \).
To find the product \( AB \), multiply matrix A by matrix B using the rule for matrix multiplication: the element in the \( i^{th} \) row and \( j^{th} \) column of \( AB \) is calculated as \( (AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} \). Perform this for all four elements to get the resulting 2x2 matrix.
Similarly, to find the product \( BA \), multiply matrix B by matrix A using the same matrix multiplication rule: \( (BA)_{ij} = b_{i1}a_{1j} + b_{i2}a_{2j} \). Calculate each element to form the resulting 2x2 matrix.
Remember that matrix multiplication is generally not commutative, so \( AB \) and \( BA \) may not be the same matrix. Carefully compute each product separately.
After computing both products, write down the resulting matrices \( AB \) and \( BA \) explicitly, showing all intermediate multiplication and addition steps for clarity.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
6mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Matrix Multiplication
Matrix multiplication involves combining two matrices by taking the dot product of rows from the first matrix with columns from the second. For two 2x2 matrices, each element of the product is calculated by multiplying corresponding elements and summing them. The order of multiplication matters, so AB and BA can yield different results.
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Properties of 2x2 Matrices
A 2x2 matrix has four elements arranged in two rows and two columns. Operations like addition, multiplication, and finding inverses are defined specifically for this size. Understanding the structure helps in performing calculations accurately and recognizing special properties like the identity matrix.
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Multiplicative Identity Matrix
The multiplicative identity for 2x2 matrices is the identity matrix, which has ones on the diagonal and zeros elsewhere. Multiplying any 2x2 matrix by the identity matrix leaves it unchanged. Recognizing this matrix helps in understanding the role of matrix B as a multiplicative element.
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