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.