d. If A is an invertible nxn matrix, then the equation Ax = b is consistent for each b in R". O A. False; the matrix A is invertible if and only if A is row equivalent to the identity matrix, and not every matrix A satisfying Ax = b is row equivalent to the identity matrix. O B. True; since A is invertible, Ab=x for all x in R^. Multiply both sides by A and the result is Ax = b. O C. False; the matrix A satisfies Ax = b if and only if A is row equivalent to the identity matrix, and not every matrix that is row equivalent to the identity matrix is invertible. O D. True; since A is invertible, Ab exists for all b in R^. Define x =A 'b. Then Ax = b. e. Each elementary matrix is invertible. O A. True; since each elementary matrix corresponds to a row operation, and every row operation is reversible, every elementary matrix has an inverse matrix. O B. False; it is possible to perform row operations on an nxn matrix that do not result in the identity matrix. Therefore, not every elementary matrix is invertible. OC. False; every matrix that is not invertible can be written as a product of elementary matrices. At least one of those elementary matrices is not invertible. O D. True; since every invertible matrix is a product of elementary matrices, every elementary matrix must be invertible.

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.