Linked Questions

89
votes
3answers
19k views

Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
11
votes
3answers
795 views

Why is the matrix multiplication defined as it is? [duplicate]

Matrix multiplication is defined as: Let $A$ be a $n \times m$ matrix and $B$ a $m\times p$ matrix, the product $AB$ is defined as a matrix of size $n\times p$ such that $(AB)_i,_j = \sum\limits_{k=...
45
votes
2answers
24k views

Matrix multiplication: interpreting and understanding the process

I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ It ...
5
votes
2answers
883 views

Why is Matrix Multiplication Not Defined Like This? [duplicate]

I'm sure everyone already thought about this at least one time. Why matrix multiplication is not defined the way showed below? $$\left( \begin{array}{ccc} a_{11} & a_{12} & \ldots \\ a_{21} &...
3
votes
2answers
87 views

How to find a basis for $W = \{A \in \mathbb{M}^{\mathbb{R}}_{3x3} \mid AB = 0\}$

$B = \begin{bmatrix} 1 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 4 & 1 \end{bmatrix}$ and I need to find a basis for $W = \{A \in \mathbb{M}^{\mathbb{R}}_{3x3} \mid AB = 0\}$ ....
3
votes
2answers
1k views

Why is matrix multiplication defined a certain way? [duplicate]

Why is it that when multiplying a (1x3) by (3x1) matrix, you get a (1x1) matrix, but when multiplying a (3x1) matrix by a (1x3) matrix, you get a (3x3) matrix? Why is matrix multiplication defined ...
3
votes
2answers
450 views

Matrix Multiplication - Why Rows $\cdot$ Columns = Columns?

I'm nearing the end of my first year of Calculus and am pretty confident in the parts of it I've learned, yet I still don't have a good understanding of matrices, which seem like they should be easier ...
2
votes
2answers
97 views

Calculating the determinant as a product without making any calculations

My problem is on the specific determinant. $$\det \begin{pmatrix} na_1+b_1 & na_2+b_2 & na_3+b_3 \\ nb_1+c_1 & nb_2+c_2 & nb_3+c_3 \\ nc_1+a_1 & nc_2+a_2 & nc_3+a_3 \...
2
votes
2answers
5k views

Definition of matrix-vector multiplication

I have just learned about matrix-vector multiplication.Is there a particular reason why we multiply a matrix by a column vector instead of a row vector? For example $Ax = \begin{bmatrix}a&b\\c&...
1
vote
2answers
529 views

Is “basis times square matrix” a new basis?

Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis for $V$. Now we take an arbitrary square matrix $S \neq 0$. $BS$ is just a linear combination of $B$. Thus $BS$ should be a new ...
6
votes
1answer
249 views

Morphism between matrices and linear equations

I'm currently a beginner at linear algebra. So, in some books I see authors start defining linear equations and then they define matrices and, supposedly, the definition of associative matrix is to ...
5
votes
1answer
1k views

What does matrix multiplication have to do with scalar multiplication?

Why are matrix and scalar multiplication denoted the same way and treated as the same operation in standard mathematical notation? This is always a source of confusion for me because they have ...
2
votes
1answer
223 views

show that $((GL(2,\mathbb{R}),\bullet)$ is a group

Let $G(2,\mathbb R)=\{\text{All invertible }2 \times 2\text{ matrices over }\mathbb{R}\}$. Then i want to show that $((G(2,\mathbb{R}),\bullet)$ is a group, where $\bullet$ is multiplication of ...
8
votes
0answers
222 views

What was the original motivation for matrix multiplication? [duplicate]

When I took linear algebra class in my freshman year, the multiplication operation for matrices was defined without any apparent motivation. Given an $m$-times-$n$ matrix $A$ and an $n$-times-$p$ ...
2
votes
0answers
48 views

Multiplication of matrices [duplicate]

When we add two matrices we just simply add the corresponding elements but when we multiply two matrices there is a much more complex process.Why does it happens?

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