The Matrix Trivia Quiz

Answer: A has m rows and n columns

The dimension of a matrix is always stated as rows x columns. The numbers m and n are always positive integers (although the actual numbers in the matrix can take on any values). Matrices are usually referred to by a single capital letter - this quiz will usually refer to the matrix A.

Answer: square matrix

A square matrix has the same number of rows as it has columns - making it have a square shape. The size of the square is usually indicated by saying that A is a square matrix of order m, so a 3x3 matrix is of order 3.

Answer: both M and N are vectors

M is called a column vector, as it only has four numbers lined up in a single column. N is called a row vector, as it has four numbers arrayed in a single row.

Answer: A = [1 3]; B = [4 -2]

In order for two matrices to have corresponding elements, they must be of the same dimension. A = [1 3] and B = [4 -2] are both 1x2 matrices, so they can be added. A+B = [5 1], since 1+4=5 and 3-2=1.

None of the other pairs have the same shape, so they cannot be added.

Answer: all elements are 0

Addition is performed by adding corresponding elements. Adding 0 to each element of matrix A does not change its value, so the result is the same as the original. A + 0 = A, where 0 stands for the Additive Identity Matrix. This corresponds to the process of adding 0 for real numbers.

Answer: the number of columns in A equals the number of rows in B

The cross product, also sometimes called the vector product, of two matrices is obtained by matching the elements in the first row of A with the elements in the first column of B, multiplying the number pairs and adding the products. This gives the element in the first column of the first row in the product matrix. The process is repeated, matching A's first row with B's columns until the end is reached, with the answers forming the first row of the product. Then the second row is matched with the columns of B in order to produce the second row of the product. And so on. This process only works if the number of columns in A equals the number of rows in B.

The final answer is not necessarily the same shape as either of the starting matrices. If an mxp matrix is crossed with a pxn matrix, the result is an mxn matrix.

Crossing a row vector with a column vector produces a rectangular matrix. When this is applied to practical situations in which each of the two original vectors has three numbers corresponding to three-dimensional coordinates of the vector quantity being considered, it is a useful shortcut used in high school Physics classes that the magnitude of the product will be equal to the magnitudes of the two original vectors multiplied by the sine of the angle between them; its direction will be perpendicular to both of the original vectors. Remember this for the joke at the end of the quiz!

Answer: multiplicative commutavity, A x B = B x A

Without going into complex calculations, consider just the dimensions of A x B and B x A if A is a 3x1 matrix and B is a 1x3 matrix. A x B will be a 3x3 matrix, while B x A will be a 1x1 matrix! They cannot possibly be the same. If one of the matrices is the Multiplicative Identity Matrix or the Zero Matrix, or if B is the multiplicative inverse of A, then the multiplication is commutative.

The other three field properties apply to matrices as well as to real numbers.

Answer: A x A^-1 gives a unit matrix

When you multiply a matrix by its inverse, the result is the Multiplicative Identity Matrix, also called the unit matrix for that order of matrices. This matrix has the number 1 entered for all elements of its leading diagonal (top left to bottom right of the square) and 0 for all other elements. When you multiply any square matrix by the appropriately-sized unit matrix, the result is the original matrix. This is the matrix equivalent of multiplying by 1.

Actually finding the inverse of a matrix can involve some nasty calculations, which I won't detail here. If you're interested, a course in Linear Algebra will give you the chance to explore more!

Answer: scalar product

The scalar product, also called the dot product, of two vectors is calculated by multiplying the corresponding elements together, then adding the products. The answer is a number. For example, [1 3].[2 1] = 5, because 1x2 + 3x1 = 5.

Answer: It isn't defined as a matrix operation

Division of matrices is not defined. Instead of dividing A by B, it is necessary to find the inverse matrix of B, B^-1, then multiply A by B^-1. Of course, this is only possible if B has an inverse. Matrices which do not have an inverse are said to be singular, and (despite that tag) are not uncommon.