Suppose matrix product ab is defined. (a) if a is 4 x 7 and b is a column matrix, give the dimensions of b and ab. [4 marks] (b) if a is the identity matrix and b is 5 x 5, what size is a? [4 marks] (c) if a is 4 x 5 and ab is 4 x 7, what size is b?

(a) Order of matrix B is 7 × 1 and order of matrix AB is 4 × 1.

(b) Order of matrix A is 5 × 5.

(c) The order of matrix B is 5 × 7.

Step-by-step explanation:

The product of two matrices is possible if and only if column of first matrix is equal to row of second matrix.

If A is an n × m matrix and B is an m × o matrix, their matrix product AB is an n × p matrix,

(a)

It is given that order of matrix A is 4 x 7.

B is a column matrix. It means the number of column in matrix B is 1.

Let the order of matrix B = n × 1

Product of matrices A and B is possible if and only if column of matrix A is equal to row of matrix B.

Order of matrix B is 7 × 1

Order of matrix AB is 4 × 1.

(b)

Order of matrix B = 5 × 5

It is given that A is the identity matrix.

Identity matrix is a square matrix having 1s on the main diagonal, and 0s everywhere else.

So, order of matrix A is equal to order of matrix B.

Order of matrix A = 5 × 5

Therefore order of matrix A is 5 × 5.

(c)

Order of matrix A = 4 × 5

Order of matrix AB = 4 × 7

Let Order of matrix B = n × m

Product of A and B are possible if and only if n=5.

On comparing both sides, we get

Therefore the order of matrix B is 5 × 7.

answer: all of the odd numbers were opened

the first person was the one that was opening all of the lockers. since one is odd, this causes all of the odd number lockers to be opened in the end.

system of linear equation

step-by-step explanation:

as given in the data

a= 20 , b= -12

let us suppose that it is a straight line equation

now we know the formula for the system of linear equation

b= ma + c )

here m is slope of a straight line which is equal to formula

m=   (ii)

and here c is a constant

now putting the value of a and b in equation (ii)

m=

m=

putting the value of a,b and m in equation (i) gives us

-12 = + c

-12=-12 + c

adding 12 to both sides

-12 +12 = -12 +12 +c

0 =c

or c=0

as the value of the constant is zero which gives us that it is a linear equation and this order pair satisfies the system of linear equations.