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Sarah Mitchell walks through inverting a three by three matrix by the adjoint method for MATSEC Advanced Matriculation Pure Mathematics. She takes the determinant first and stops on a singular matrix, builds nine minors, applies the plus minus plus checkerboard, transposes the cofactor matrix into the adjoint, and keeps one over the determinant outside as a scalar factor. Two worked problems cover a full inverse with determinant thirteen and a value of k that kills the inverse. Lesson 135 of 158 in the MATSEC AM Pure Mathematics Complete Prep Course audio course from ExamPrepRise. Watch the video course: https://exampreprise.com/courses/mt-matsec-am-pure-mathematics-fundamentals Free practice questions: https://exampreprise.com/practice/mt-matsec-am-pure-mathematics Always confirm the current test plan, eligibility rules, fees and scheduling details with the official test provider before you register.

The lesson shows how to expand a three by three determinant along any row or column, multiplying each entry by its two by two minor and attaching checkerboard signs. It explains why choosing the line with the most zeros cuts the arithmetic, and how a zero determinant marks a matrix as singular rather than invertible. Two worked problems cover a row with two zeros and solving for a singular parameter. Lesson 134 of 158 in the MATSEC AM Pure Mathematics Complete Prep Course audio course from ExamPrepRise. Watch the video course: https://exampreprise.com/courses/mt-matsec-am-pure-mathematics-fundamentals Free practice questions: https://exampreprise.com/practice/mt-matsec-am-pure-mathematics Always confirm the current test plan, eligibility rules, fees and scheduling details with the official test provider before you register.

Sarah Mitchell shows how to locate tangents parallel and perpendicular to the initial line on a polar curve. She derives the two product rule expressions for the derivative of r sine theta and r cosine theta, explains why a horizontal tangent needs one zero and the other non zero, handles the indeterminate case at the pole, and works the cardioid r equals a times one plus cosine theta twice. Lesson 133 of 158 in the MATSEC AM Pure Mathematics Complete Prep Course audio course from ExamPrepRise. Watch the video course: https://exampreprise.com/courses/mt-matsec-am-pure-mathematics-fundamentals Free practice questions: https://exampreprise.com/practice/mt-matsec-am-pure-mathematics Always confirm the current test plan, eligibility rules, fees and scheduling details with the official test provider before you register.

Derives the polar area formula, one half the integral of r squared with respect to theta from alpha to beta, and shows how to pick limits that sweep a region exactly once. Covers the double angle substitutions, symmetry shortcuts, outer squared minus inner squared between two curves, intersections by equating radii, and the hidden intersection at the pole. Worked examples cover one petal of a four petal rose, and a circle minus cardioid area of pi. Lesson 132 of 158 in the MATSEC AM Pure Mathematics Complete Prep Course audio course from ExamPrepRise. Watch the video course: https://exampreprise.com/courses/mt-matsec-am-pure-mathematics-fundamentals Free practice questions: https://exampreprise.com/practice/mt-matsec-am-pure-mathematics Always confirm the current test plan, eligibility rules, fees and scheduling details with the official test provider before you register.

Sarah Mitchell shows how to move a curve between Cartesian and polar form using x equals r cos theta, y equals r sin theta and x squared plus y squared equals r squared. She derives the polar equation of a general line, the special cases x equals k and theta equals a constant, and the multiply-by-r trick that turns r equals two a cos theta into a circle of radius a centred at a comma zero, plus the pole and extra-branch traps. Lesson 131 of 158 in the MATSEC AM Pure Mathematics Complete Prep Course audio course from ExamPrepRise. Watch the video course: https://exampreprise.com/courses/mt-matsec-am-pure-mathematics-fundamentals Free practice questions: https://exampreprise.com/practice/mt-matsec-am-pure-mathematics Always confirm the current test plan, eligibility rules, fees and scheduling details with the official test provider before you register.