1. A matrix A over R is called symmetric if At = A and skew-symmetric if At = −
2. A matrix A is said to be orthogonal if AAt = AtA =

Examples

1.   Let
and
Then A is a symmetric matrix and B is a skew-symmetric matrix.
2. Let

Then A is an orthogonal matrix.
3. Let A = [aij ] be an n×n matrix with

Then An = 0 and Aℓ 6= 0 for 1 ≤ ℓ ≤n − 1. The matrices A for which a positive integer k exists such that Ak = 0 are called nilpotent The least positive integer k for which Ak = 0 is called the order of nilpotency.
4. Let

Then A2 = The matrices that satisfy the condition that A2 = A are called idempotent matrices.

1. Show that for any square matrix A, S = 2 (A + At) is symmetric, T = 1/2(A+At ) skew-symmetric, and A = S +
2. Show that the product of two lower triangular matrices is a lower triangular matrix. A similar statement holds for upper triangular matrices.
3. Let A and B be symmetric matrices. Show that AB is symmetric if and only if AB =
4. Show that the diagonal entries of a skew-symmetric matrix are zero.
5. Let A,B be skew-symmetric matrices with AB = Is the matrix AB symmetric or skew-symmetric?
6. Let A be a symmetric matrix of order n with A2 = 0. Is it necessarily true that A = 0?
7. Let A be a nilpotent matrix. Show that there exists a matrix B such that B(I + A) = I = (I + A)

Submatrix of a Matrix
For example, if

a few submatrices of A are

But the matrices

Miscellaneous Exercises
Let

Geometrically interpret y = Ax
SOME MORE SPECIAL MATRICES

• Compose the two transformations to express x1, x2 in terms of z1, z2.
• If xt = [x1, x2], yt = [y1, y2] and zt = [z1, z2] then find matrices A,B and C such that x = Ay, y = Bz and x = Cz.
• Is C = AB?
• For a square matrix A of order n, we define trace of A, denoted by tr (A) as tr (A) = a11 + a22 + · · · ann.
Then for two square matrices, A and B of the same order, show the following:
(a) tr (A + B) = tr (A) + tr (B).
(b) tr (AB) = tr (BA).
• Show that, there do not exist matrices A and B such that AB − BA = cIn for any c 6= 0.
• Let A and B be two m × n matrices and let x be an n × 1 column vector.
(a) Prove that if Ax = 0 for all x, then A is the zero matrix.
(b) Prove that if Ax = Bx for all x, then A = B.
• Let A be an n × n matrix such that AB = BA for all n × n matrices B. Show that A = αI for some α ∈
• Let

Show that there exist infinitely many matrices B such that BA = I2. Also, show that there does not exist any matrix C such that AC = I3.
Block Matrices
Let A be an n × m matrix and B be an m × p matrix. Suppose r < m. Then, we can decompose the matrices A and B as A = [P Q] and

where P has order n × r and H has order r × p. That is, the matrices P and Q are submatrices of A and P consists of the first r columns of A and Q consists of the last m − r columns of A. Similarly, H and K are submatrices of B and H consists of the first r rows of B and K consists of the last m− r rows of B. We now prove the following important theorem.
Theorem

Proof. First note that the matrices PH and QK are each of order n × p. The matrix products PH and QK are valid as the order of the matrices P,H,Q and K are respectively, n ×r, r ×p, n ×(m− r) and (m−r)×p. Let P = [Pij ], Q = [Qij ], H = [Hij ], and K = [kij ]. Then, for 1 ≤ i ≤ n and 1 ≤ j ≤ p, we have

Theorem@ is very useful due to the following reasons:

1. The order of the matrices P,Q,H and K are smaller than that of A or B.
2. It may be possible to block the matrix in such a way that a few blocks are either identity matrices or zero matrices. In this case, it may be easy to handle the matrix product using the block form.
3. Or when we want to prove results using induction, then we may assume the result for r × r submatrices and then look for (r + 1) × (r + 1) submatrices, etc.

For example, if