How do you know if a matrix multiplication is possible?

You can only multiply two matrices if their dimensions are compatible , which means the number of columns in the first matrix is the same as the number of rows in the second matrix. If A=[aij] is an m×n matrix and B=[bij] is an n×p matrix, the product AB is an m×p matrix.

How do you prove a matrix is closed under multiplication?

If we can multiply two matrices, the product is a matrix: matrices are closed under multiplication. As noted above, matrix multiplication, like that of numbers, is associative, that is, (AB)C = A(BC). Unlike numbers, matrix multiplication is not generally commutative (although some pairs of matrices do commute).

Is the set of 2×2 matrices A group under multiplication?

The set of all 2 x 2 matrices with real entries under matrix multiplication is NOT a group.

How do you know if vectors are closed under addition?

So a set is closed under addition if the sum of any two elements in the set is also in the set. For example, the real numbers R have a standard binary operation called addition (the familiar one). Then the set of integers Z is closed under addition because the sum of any two integers is an integer.

Is matrices a group under matrix multiplication?

groups under multiplication. The set Mn(R) of all n × n matrices under matrix multiplication is not a group. The n × n matrix with all entries 0 has no inverse. The set GL(n,R) of all n × n invertible matrices with matrix multiplication is a non-commutative group!

What makes multiplying matrices undefined?

Matrix Multiplication Defined. Just as with adding matrices, the sizes of the matrices matter when we are multiplying. If, using the above matrices, B had had only two rows, its columns would have been too short to multiply against the rows of A. Then “AB” would not have existed; the product would have been “undefined” …

Why are some matrices undefined?

The product of two matrices is undefined whenever the rows of the first matrix (reading right to left) do not match the column of the second matrix.