Inverse of a Matrix

The inverse of a square non-singular matrix A is the matrix of cofactors of A divided by the determinant of A. A cofactor of an element of A is the determinant of the matrix which results from crossing out the row and column of A that contains the element, and which is then multiplied by -1 to the power of i+j, where i is the row and j is the column. Non-singular means that the determinant of A doesn't equal zero.

What is the inverse of the matrix A given below? Some of the cofactors of A and the determinant of A have been computed for you.

Trace of a Matrix

The trace of a matrix is the sum of the diagonal elements of the matrix.

Let U be the 3x3 unit matrix:

1 0 0
0 1 0
0 0 1

The trace of U is Tr(U) = 3.

Does Tr(5U) = 15 ?

Determinant of a Matrix

Let U be the 3x3 unit matrix:

1 0 0
0 1 0
0 0 1

The determinant of U is Det(U) = 1.

Does Det(5U) = 5 ?

Square Root

Suppose M and m are positive numbers, with M greater than m. Does the square root of the square of  (m - M):

sqrt((m - M)(m - M))

equal

(m - M) ?

Euclid's Proof of Pythagorean Theorem

Let the yellow triangle be a right triangle. Show that the sum of the areas of the blue and green squares equals the area of the red square. Verify the following:

• Notice that triangles ABD and CDF are congruent (identical)
• Area of triangle CDF = one-half the area of the blue square
• Area of triangle ABD = one-half the area of rectangle AGED
• Therefore, area of the blue square = area of rectangle AGED
• Surmise that a similar analysis will show that the area of the green square = area of rectangle GHFE
• And therefore, that the sum of the areas of the blue and green squares equals the area of the red square

Since the area of each square = the square of the corresponding side of the yellow triangle, the theorem of Pythagoras is proved.

Intersecting Circles: Longest Line

Show that line BAC is longest when lines BD and CD pass through the centers of the circles. Also show that line BAC is perpendicular to line AD when this happens.

• Notice that angles B and C don't change as line BAC is rotated about point A. (Investigate some of the posts below with the large yellow circles.)
• This means that all triangles BDC are similar (have the same shape, but differ in size). So all sides of triangle BDC must maximize at the same time.
• Note also that the line between the centers of the circles is perpendicular to line AD, because the two circles are symmetric about the horizontal line drawn through their centers.

Circles in Circles

If the radius of the small circle is 1, and the radius of the large circle is 3, what is the radius of the still larger circle that has as arcs those curves which are tangent to the small circle?

Intersecting Circles

Two circles of radius 1 intersect in such a way that the perimeter of each circle passes through the center of the other circle. Show that the area of intersection (the GREEN area) has area: 2*pi/3 - sqrt(3)/2.

• Draw an equilateral triangle with vertices at the centers of the circles and at the point of intersection of the two circles at the top of the figure
• Calculate the area of the triangle: 1/2 * base * height = 1/2 * sqrt(3)/2
• Area of corresponding sector of the circle = pi/6
• Difference between areas = pi/6 - sqrt(3)/4
• Add this difference to the area of the sector of the circle. The result is: pi/3 - sqrt(3)/4
• Multiply by 2 to include the lower half of the green area

Show: Angle B = 2 * Angle A

Show that angle B at the center of the circle is twice as large as angle A.

Show: Angle F = 2 * Angle E

Show that angle F at the center of the circle is twice as large as angle E. (Hint: Use the result of the above post: Draw two figures like the figure in the above post, but in the same circle, and then add or subtract angles.)

Inscribed Quadrilateral

Prove: A + C = B + D = 180 degrees. (Hint: Draw a line from B to the center of the circle, and another line from the center of the circle to D. Then use the result of the above post. The sum of the central angles must equal 360 degrees. This will yield A + C = 180 degrees. Repeat for angles B and D.)

Japanese Temple Problem from 1844

Show that the area of the red triangle = the area of the red square. (Hint: Check out the figure here. Note that the area of the red triangle = area of trapezoid minus the area of the two triangles within the trapezoid. The area of the trapezoid = (a+b)*(1/2)*(2a+2b).)

Diameter of Circle inscribed in Trapezoid = ?

Find the diameter of the circle inscribed in the trapezoid. (Hint: The two tangents from a given point to a given circle have the same length. The length of the slanted sides of the trapezoid should therefore be immediately obvious.)

Area of Triangle = A; Area of Ellipse = B = ?

Shown are four identical rectangles, as well as a triangle with area A, and an ellipse with area B. What does area B equal in terms of area A? No pencil or paper is needed. (Hint: Area of an ellipse = pi*(semimajor axis)*(semiminor axis.))

Odd Pair of Letters

Which is the odd pair of letters?
BY, HS, EV, JR, GT

Odd Word Out

Which word is the odd one out?
BLITHERING CATHEDRAL WAREHOUSE
PROMETHEAN STEPMOTHER

Rearrangement

Rearrange the letters in NEW DOOR to form one word.

Sequence 5

What number comes next in the sequence?
1 11 21 1211 111221 ?

Die Throws

With a standard six-sided die, how many throws are required on average before each of the six numbers has landed faceup?
(a) 10 (b) 15 (c) 20 (d) 25

Moving Digits

Move one digit to a new position so that the equation below is
correct. (Moving signs is not allowed.)
62 – 63 = 1

Sequence 4

What is the next number in the sequence?
1 4 7 12 23 42 ?

Sequence 3

What is the next number in the series?
77 49 36 18 ?

Passenger Seats

The cost of hiring a private rail carriage is shared equally by all the passengers. The carriage has seats for forty passengers and the total bill amounts to $70.37. How many passenger seats are not occupied?

Sequence 2

What number replaces the question mark?
1 4 3 11 15 13 17 ?

Sequence 1

What number comes next in the sequence?
4 6 5 4 2 3 ?

Two Boats

Two boats travelling at a constant speed start moving at the same instant from opposite sides of the Hudson River, one going from New York City to Jersey City, and the other from Jersey City to New York City. They pass one another at a point 720 yards from the New York shore.

After arriving at their respective destinations, each boat spends precisely 10 minutes at the opposite shore to change passengers before switching directions. On the return trip, the two boats meet at a point 400 yards from the Jersey shore.

After arriving at their respective destinations, each boat again spends precisely 10 minutes at the opposite shore to change passengers before switching directions. How far from the New York shore will the two boats be when they meet for the third time?

(Hint: Find the width of the river. When the two boats meet for the first time, they will have travelled a total distance of one width of the river. How many widths will they have travelled by the time they meet the second time? (The answer is NOT two widths.) By the time thet meet the third time? Do the 10 minutes even matter?)