4. Determinants

9 topics3 h

Chapter 4 of NCERT Mathematics for Class 12: Determinants. Three sections — One number for a square matrix, Minors, cofactors and the adjoint and Solving a system of linear equations. Nine videos, 3 h in all.

Worked answers to the 61 exercise questions in this chapter: Exercise 4.1 · Exercise 4.2 · Exercise 4.3 · Exercise 4.4 · Exercise 4.5 · Miscellaneous Exercise

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Part 1

One number for a square matrix

17 minWhy only square arrays get this number, and what it settles about a linear systemTwo straight lines either cross once, never meet, or lie on top of each other - and four coefficients settle which, before you solve anything. Two multiplications and a subtraction. Here is that number, measured against five rival rules that each fail somewhere, and against the one rival that passes and is still wrong.17 min · 14 parts22 minOrders one, two and three, and expanding along a chosen rowOrder three is not a new formula. It is order two, used three times - and order two is order one, used twice. Get that and you also get a freedom: six different lines to expand along, all giving the same number, so you may always take the one with the most zeros in it. Learn it as a formula instead and you throw that away.22 min · 16 parts21 minArea of a triangle from its vertices, and why three collinear points give zeroThe area of a triangle from its three corners is an expression nobody remembers and an array anybody can. But the rewrite is not only a memory aid: it turns a question about geometry - are these three points in a line? - into one number you can work out and look at. And the column of ones that makes it work is not padding.21 min · 15 parts

Part 2

Minors, cofactors and the adjoint

22 minDeleting a row and a column: minors, and the sign that turns one into a cofactorNothing new is computed here. The smaller determinant left when you strike out a row and a column was already there; so was the sign built from a position's two subscripts. All that happens is that both get names - and an eighteen-slot expansion becomes three products added.22 min · 15 parts20 minWhy cofactors borrowed from the wrong row always sum to zeroPair a line's entries with its own cofactors and you get the determinant. Pair them with a DIFFERENT line's cofactors and you get zero - every array, every time. The reason is that the mismatched sum is secretly the expansion of an array with one row written twice, and the last step of that argument is a fact most courses use here without ever stating it.20 min · 15 parts17 minBuilding the adjoint, and why it multiplies back to a scalar times the identityThe adjoint looks like an arbitrary recipe: replace every entry by its cofactor, then flip the whole array about its main diagonal. The flip is the part nobody explains - and it is the entire mechanism. It is what puts a ROW of cofactors where the multiplication needs a COLUMN, which is why the product comes out as the determinant down the diagonal and nothing anywhere else.17 min · 18 parts19 minSingular against non-singular, and the one test an inverse has to pass19 min · 18 parts

Part 3

Solving a system of linear equations

19 minPacking three equations into AX = B and reading the solution off the inverseThree equations in three unknowns become one equation in one unknown the moment the unknown is allowed to be a column. Everything after that is the ordinary business of undoing a multiplication - except that the thing being undone is an array, so the undoing is an inverse, and the side it goes on is decided by the packing rather than by a rule to be memorised.19 min · 17 parts17 minConsistent or inconsistent: what a vanishing determinant does and does not decideA vanishing determinant is not a verdict. It is the loss of a tool. The inverse disappears, the method stops working, and whether the system has any answer at all is still wide open. There is a second test - push the right-hand column through the adjoint - and the honest lesson is that it is not a verdict either: it settles one case cleanly and leaves the other genuinely undecided.17 min · 17 parts

What you will be able to do

  • State what a determinant is: a rule that takes a square array and returns one number, and does not accept anything else
  • Evaluate a determinant of order one, and say why the case is worth stating
  • Write the earlier-class area expression and rearrange it into a three-row determinant with a column of ones
  • State what the minor of an entry is, and produce it by deleting the entry's own row and its own column
  • State the correct pairing that produces a determinant, and contrast it with the mismatched pairing
  • Build the adjoint of a square array in two explicit moves and say what each move does
  • State the two definitions this section introduces and place any given array on one side of the line by one computation
  • Rewrite a system of two or three linear equations as one matrix equation, and name the three arrays involved
  • State the two outcomes a system can be sorted into, before computing anything

What this chapter assumes you already know

  • Matrices from Chapter 3: entries, rows, columns, and the order of an array
  • Which arrays count as square, and what the diagonal entries are
  • Writing a pair of simultaneous equations as one matrix equation
  • Solving two linear equations in two unknowns by elimination, from an earlier class
  • The idea that two straight lines may cross once, never, or everywhere
  • Double-subscript notation, and reading the row index before the column index
  • The absolute value of a real number, and the bars it is written with
  • What a determinant is, and that only square arrays have one

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "The bars mean absolute value, so a determinant is never negative."
  • "Every matrix has a determinant."
  • "The determinant tells you the solution."
  • "If the number is zero there is no solution."
  • "Determinant and matrix are two words for the same thing."
  • "The order of a matrix is the order you write its entries in."
  • "Complex entries are examinable in this chapter."
  • "The chapter is going to give me a list of properties to memorise."