Math Solutions

Compute det([[1,2,3],[4,5,6],[7,8,0]]) by cofactor expansion along row 1. Show each minor determinant.

Compute det([[3,7],[1,5]]) and interpret it as the signed area of the parallelogram formed by the column vectors.

Find the inverse of A = [[2,1],[5,3]] using A⁻¹ = (1/det A) · adj(A). Verify A · A⁻¹ = I.

Find the transpose of A = [[1,2,3],[4,5,6]]. Explain how the transpose swaps rows with columns and list the key properties.

Compute AB for A = [[1,2],[3,4]] and B = [[5,6],[7,8]]. Show each entry as a row-column dot product.

For A = [[1,2],[3,4]] and B = [[5,6],[7,8]], compute A + B, A − B, and 3A. Explain why these operations are element-wise.

Transform the matrix [[1,2,3],[4,5,6],[7,8,10]] to reduced row echelon form and identify the pivots.

Solve the system 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3 using row operations on the augmented matrix.

A $1,000 face-value bond pays a 5% annual coupon for 10 years. Market interest rates are 6%. Find the bond price.

Pay $1,000 today and receive $400/year for 3 years. Find the IRR (the rate where NPV = 0) and show the NPV-vs-rate curve.

A credit card has an 18% APR compounded daily. Find the APY and explain why it exceeds the APR.

A $300,000 house with 20% down, financed over 30 years at 3.5% annual interest. Find the monthly payment, total interest paid, and amortization split.

Would you rather have $1,000 today or $1,200 in 3 years at 8% opportunity cost? Compare the two options by their present values.

You need $100,000 in 10 years. How much must you deposit at the end of each quarter at 5% annual interest compounded quarterly?

An investment costs $50,000 today and returns $15,000/year for 5 years. Compute NPV at a 10% discount rate and decide if it is worth funding.

Fixed costs of $10,000/month, variable cost $15/unit, price $25/unit. Find the break-even quantity and graph revenue vs. total cost.

A machine costs $50,000 with a 10-year useful life and $5,000 salvage value. Compare straight-line depreciation with double-declining balance.

Find the monthly payment for a $25,000 car loan at 5.5% annual interest for 5 years using PMT = P·r(1+r)^n / [(1+r)^n − 1].

A nominal rate of 12% compounded monthly. Find the effective annual rate (EAR) and compare to other compounding frequencies.

At 6% annual interest, roughly how long will money take to double? Verify with the exact log formula and assess the approximation error.