MAT 242 Chap.3 Matrix Arithmetic and Matrix Multiplication
Matrix Arithmetic and Matrix Multiplication
Matrices become objects
This chapter treats a matrix as a thing you can compute with. An m × n matrix has m rows and n columns; its entry in row i and column j is written with two subscripts. Equality demands identical dimensions and identical entries. Addition works entry by entry between matrices of the same size, and a scalar multiplies every entry.
These operations follow the algebra you expect, and each size has its own zero matrix.
The product, and why it is defined this way
Multiplication is the operation that surprises students. AB exists only when A has as many columns as B has rows, and the entry in row i, column j of AB is the dot product of row i of A with column j of B.
The order matters: AB and BA can differ in shape or in entries, and one of them may not exist at all. A nonzero matrix can even square to zero, and a common factor cannot simply be cancelled. Associativity and the distributive laws do survive.
The payoff
Multiply a coefficient matrix by a column of variables and each row reproduces the left side of an equation.
A whole linear system therefore becomes one matrix equation, AX = B, and solving it becomes the search for an identity matrix and an inverse that the next chapter completes.
The paid chapter solves a matrix equation by matching entries, builds a product entry by entry with a figure, and translates systems into matrix form and back.
Test 1 is taken without a calculator, so products and sums are practised by hand, with a size check before any arithmetic starts.
Solving equations with matrices in them
A typical question hides unknowns inside matrices and asks for all values that make an equation true.
The method is to simplify each side into a single matrix and then compare entries, which turns the problem into ordinary equations. When one of those equations is quadratic it may offer two candidates, and the other entries decide which survive, so every candidate must be tested in every entry.
Sample Test 1 questions also ask only for the system of equations that a product produces, without solving it, so read the command before starting.
Properties you may use without proof
The notes list the algebra that does hold for products: associativity, both distributive laws, scalars moving freely through a product, and zero matrices absorbing products. What fails is commutativity and cancellation.
When simplifying an expression such as A(B + C) − AB, the distributive law gives AC directly; when an expression needs BA but you have AB, the two cannot be exchanged without more information.
What this chapter covers
- 01
Entries, rows, columns and dimensions
- 02
Matrix equality, addition and scalar multiples
- 03
When a product is defined
- 04
Computing AB entry by entry
- 05
Writing a system as AX = B
Multiply two 2 × 2 matrices
- 2Row 1 with each column: 1·3 + 2·2 = 7 and 1·1 + 2·4 = 9.
- 2Row 2 with each column: 0·3 + (−1)·2 = −2 and 0·1 + (−1)·4 = −4.
Key terms
- Dimensions
- Dimensions record a matrix's size as rows by columns, written m × n.
- Scalar Multiple
- A scalar multiple is the matrix obtained by multiplying every entry by the same real number.
- Identity Matrix
- The identity matrix is square with 1s on the main diagonal and 0s elsewhere, so multiplying by it changes nothing.
Matrix Arithmetic and Matrix Multiplication FAQ
How can I tell quickly whether two matrices can be multiplied?
Write their sizes side by side. The inner numbers must agree, the columns of the left factor against the rows of the right one, and the outer numbers give the size of the product.
If AC equals BC, can I conclude A equals B?
Not in general. Matrix factors cannot be cancelled like numbers, and a nonzero matrix can multiply to zero. Cancelling is only safe with an invertible factor on the same side.
Why does the course define matrix multiplication with rows and columns?
Because it makes a coefficient matrix times a column of variables reproduce the left sides of a linear system, so a whole system becomes the single equation AX = B.
Exam move
Before multiplying anything, write the size check. Practise with rectangular pairs, not only squares, because mismatched shapes are where exam questions hide undefined products. Mix in one question per session that asks for a product in both orders, so that checking definedness and shape becomes automatic.
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