3. Matrices

10 topics3 h

Chapter 3 of NCERT Mathematics for Class 12: Matrices. Three sections — What a matrix is, The algebra of matrices and Transpose, symmetry and inverse. Ten videos, 3 h in all.

Worked answers to the 56 exercise questions in this chapter: Exercise 3.1 · Exercise 3.2 · Exercise 3.3 · Exercise 3.4 · Miscellaneous Exercise

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Part 1

What a matrix is

19 minAn ordered rectangular array, and what its order tells you before anything elseThe same six counts, poured into a three-by-two frame and into a two-by-three frame, hold the same data and are not the same object. There are 720 ways those six counts could have filled the first frame alone and 2880 ways across every frame they fit, and exactly one of them is the arrangement that means what it says.19 min · 12 parts19 minThe named shapes — row, column, square, diagonal, scalar, identity, zeroSeven names for seven kinds of matrix arrive as a flat list, and they are not flat. Four of them are one thing unfolded in four steps: over 21459 small arrays, the number that are diagonal and not square is 0, scalar and not diagonal is 0, an identity and not scalar is 0 - while the counts the other way round are 19728, 30 and 6. The other three names never ask the array to be square at all.19 min · 13 parts22 minEquality as two demands, and why the orders have to agree before the entries are looked atTwo arrays holding exactly the same numbers, and they are not equal. Over all 6561 ordered pairs of two by two arrays built from 0, 1 and 2, only 81 are equal while 639 hold the same entries - so 558 pairs hold precisely the same values and are still unequal, nearly seven near misses for every genuine equality.22 min · 13 parts

Part 2

The algebra of matrices

21 minAddition and scalar multiplication done entry by entry, and why orders must agreeAdding two matrices and scaling one are the same kind of rule - do one small thing at every position - and their domains are opposites. Over 26 small arrays, only 292 of the 676 ordered pairs can be added at all, while all 130 of the number-and-array pairs can be scaled. That asymmetry, not the arithmetic, is the content.21 min · 13 parts21 minThe laws addition and scaling obey, and what they buy youSix laws of matrix addition and scaling, and one move that settles all six: drop to a single position, use the fact about ordinary numbers there, come back up. To show the move is doing real work, the same laws are put to entry rules where they FAIL - joining words breaks the order law in 55854 of 66064 cases, and taking midpoints breaks the grouping law in 7680 of 8192.21 min · 14 parts18 minRow into column: why multiplication needs the inner orders to matchTwo of the four numbers in a pair of orders meet in the middle, and two never meet anything. The inner pair decides whether a product exists; the outer pair IS the answer's order. Of the 256 ordered pairs of orders up to four by four, only 64 have a product - and the three rules students actually reach for get 72, 96 and 108 of them wrong.18 min · 13 parts19 minWhat multiplication keeps from ordinary algebra, and the two things it losesMatrix multiplication keeps three habits of ordinary algebra and loses two, and the two it loses are the ones your existing instincts depend on: you may not swap two factors, and you may not cancel one. Both losses are shown on two by two matrices small enough to hold in the head - and both are counted, so 'not guaranteed' does not quietly become 'never'.19 min · 13 parts

Part 3

Transpose, symmetry and inverse

17 minFlipping rows into columns, and why the transpose of a product reverses itThe transpose is one relabelling - whatever sat in row i, column j now sits in row j, column i - and three of its four properties fall straight out of that. The fourth does not: transposing a product turns the two factors around. That reversal is not a convention to memorise. Count the four order numbers once and you will see it is the only arrangement that fits.17 min · 13 parts18 minSymmetric and skew symmetric, and splitting any square matrix into one of eachTwo conditions, one minus sign apart. One of them says nothing about the diagonal; the other forces every diagonal entry to zero. And every square matrix splits into one of each - in exactly one way, with two halves that are not a choice.18 min · 14 parts20 minWhat it takes to be an inverse, and why there can only ever be oneThe definition of an inverse asks for two products, not one - and that second demand is not tidiness. It is what forces both matrices to be square, what makes the relation run both ways, and what makes the uniqueness proof work at all. Weaken it by half and all three go at once.20 min · 16 parts

What you will be able to do

  • Explain what an arrangement records that a plain list of the same numbers does not
  • Recognise each of the seven named kinds from a printed array
  • State Definition 2 as two tests applied in a fixed order
  • Add two matrices whose orders agree, working position by position
  • State each of the four laws of matrix addition and each of the two laws of scaling
  • Explain where the sum of products in a matrix product comes from, using a quantities-against-prices situation
  • Show that two products taken in opposite orders need not agree, in the case where the two have different orders and in the case where they have the same order
  • Write down the transpose of a printed matrix of any order
  • State the symmetric condition and the skew symmetric condition, each as one equation and as a condition on entries
  • State Definition 6, naming both products it demands

What this chapter assumes you already know

  • Reading a two-way table with labelled rows and columns
  • Ordered pairs, and the fact that swapping the two entries makes a different pair
  • Factorising a small whole number into a product of two factors
  • Absolute value of a real number, from Class XI
  • Subscript notation, and a rule that assigns a value to each index
  • Cartesian coordinates of a point in the plane
  • Surds, decimals and simple algebraic and trigonometric expressions as things that can sit where a number sits
  • The order of a matrix, rows before columns, from the previous topic

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.

  • "Order means columns by rows, or it does not matter which."
  • "The two notebook arrangements are the same matrix, because the data is the same."
  • "A matrix is a way of writing a number, like a determinant."
  • "Eight entries give two orders, one by eight and four by two."
  • "A single number in brackets is not really a matrix."
  • "Entries must be numbers."
  • "a sub two three and a sub three two are the same entry."
  • "Constructing a matrix from a rule needs the rule to be linear, or simple."