cosmos42 said:Homework Statement
Suppose that AB = AC for matrices A, B, and C.
Is it true that B must equal C? Prove the result or find a counterexample.
Homework Equations
Properties of matrix multiplication
The Attempt at a Solution
AC = A(D + B) = AD + AB = 0 + AB = AB ? Can someone help me understand in plain english?
Not helpfulHallsofIvy said:IF A has an inverse the AB= AC gives [itex]A^{-1}AB= A^{-1}AC[/itex] so [itex]IB= IC[/itex] so [itex]B= C[/itex] so you question becomes "does every matrix have an inverse?".
By the way, your question, as stated, is trivially false even for numbers: 0B= 0C for any B and C, it does NOT follow that B= C. To make you question at all interesting you should add "A non-zero".
Matrices sums and products are basic operations performed on matrices, which are rectangular arrays of numbers. A sum of two matrices is obtained by adding corresponding elements together, while a product of two matrices is obtained by multiplying corresponding elements and then summing the results.
Matrices sums and products are calculated by following specific rules based on the dimensions of the matrices. For sums, the matrices must have the same number of rows and columns, and the sum is simply calculated by adding corresponding elements. For products, the number of columns in the first matrix must be equal to the number of rows in the second matrix, and the product is obtained by multiplying corresponding elements and then summing the results.
Matrices sums and products have a wide range of applications in various fields such as physics, engineering, economics, and computer science. They are commonly used to solve systems of linear equations, perform transformations in computer graphics, and analyze networks and data sets.
Matrices sums and products have several important properties, including commutativity (the order of matrices in a sum or product does not affect the result), associativity (the way matrices are grouped in a sum or product does not affect the result), and distributivity (the sum of two matrices multiplied by a scalar is equal to the sum of each matrix multiplied by the scalar).
Matrices sums and products are closely related to other mathematical concepts such as vectors, determinants, and eigenvalues. They are also used in higher level mathematics such as linear algebra, which deals with the study of vector spaces and linear transformations.