2.1 Matrix Multiplication T/F

If A and B are n x n matrices, then the diagonal entries of the product matrix AB are a11b11,a22b22,...a_nn*b_nn.

False, according to the row-column rule for the (i,j)-entry of a matrix product the diagonal entry is a_i1*b_1i + a_i2*b_2i + ... + a_in*B_ni.

For any matrices A and B for which the product AB is defined, the (i,j)-entry of AB equals a_ij * b_ij.

False, according to the row-column rule for the (i,j)-entry of a matrix product the entry is a_i1*b_1j + a_i2*b_2j + ... + a_in*bnj.

For any matrices A and B for which the product AB is defined, the (i,j)-entry of AB equals the sum of the products of corresponding entries from the ith column of A and the jth row of B.

False, according to the row-column rule for the (i,j)-entry of a matrix product the entry is the sum of the products of the corresponding entries from the ith row of A and the jth column of B.

For any matrices A and B, if the products AB and BA are both defined, then AB = BA.

False, for example if A is a 2 x 3-matrix and B is a 3 x 2-matrix, then AB is a 2 x 2-matrix and BA is a 3 x 3-matrix. Since AB and BA do not have the same dimension, they are not equal.

If A and B are m x n matrices and C is an n x p matrix, then (A+B)C = AB + BC.

False, the right distributive law gives (A+B)C = AC + BC.

If A is an m x n matrix and B is an n x p matrix, then (AB)^T = A^TB^T.

False; -First check whether both the sides are defined or not -Left hand side is product of A and B i.e. (m xn) x (n x p) since inner dimensions are same so it is defined -Right hand side i.e. A^T * B^T i.e. (n x m) * (p x n) now for this product to be defined m = p which is not given hence this statement is false

If the product AB is defined and AB is a zero matrix, then either A or B is a zero matrix.

False; -To show that given statement is false it would be enough to give one example in which both A and B are non-zero but AB is zero A = [a 0;0 0] , B = [0 0; 0 d] -In the above example A is non zero , B is nonzero but AB = 0.

If A and B are matrices, then both AB and BA are defined if and only if A and B are square matrices.

False; -one example where both AB and BA are defined but A and B are not square matrices will be sufficient to say that given statement is false -and that example is say A is 2 x 3 matrix and B is 3 x 2 matrix notice here none of the A or B is square but both AB and BA are defined.

The product of two diagonal matrices is a diagonal matrix.

False; It is not given that what is the order of the diagonal matrices so if the product is not itself defined(say matrices are square of differents orders) how we can say with certainity that it will be diagonal matrix

The product of two m x n matrices is defined.

Product of two matrices of order a x b and c x d is defined when b = c here b is n and c is m so product will be defined only when m = n and hence in general product is not defined.

For any matrices A and B, if the product AB is defined, then the product BA is also defined.

Let take an example let A be 2 x 3 and B be 3 x 4 now AB is defined and is of the order 2 x 4 but BA (3 x 4)x(2 x 3) is not defined so this statement is false.

In a symmetric n x n matrix, the (i,j)- and (j,i)-entries are equal for all i= 1, 2, ... n and j = 1,2,... n and j = 1, 2, ...., n.

True, because a symmetric matrix is symmetric about its diagonal.

If A_alpha and A_beta are both 2 x 2 rotation matrices, then A_alpha * A_beta is a 2 x 2 rotation matrix.

True, the product of the two rotation matrices will give the total rotation when performing both rotations.

For any matrices A and B for which the product AB is defined, the jth column of AB equals the matrix-vector product of A and the jth column of B.

True, this is a consequence of the row-column rule for the (i,j)-entry of a matrix product.

If A, B, and C are matrices for which the product A(BC) is defined, then A(BC) = (AB)C.

True, this is the associative law of matrix multiplication.

If the product AB is defined and either A or B is a zero matrix, then AB is a zero matrix.

True; -It is given that AB is defined.And either A is zero or B is zero and in product AB each term will be sum of product of two terms one from each matrix and since it is given that each and every term of one of the matrix either A or B is zero this implies that each and every term of AB will also be zero so the given statement is true

There exists nonzero matrices A and B for which AB = BA.

True; -Statement says there exist so it is sufficient to give one example where the statement is true to say that statement is true -take A = I of order n and B be a square matrix of order n and as we know IB = BI and both(B and I) are nonzero as well hence the statement is true

If A is a square matrix, then A^2 is defined.

True; -Let A be of the order m x n then m = n as A is square matrix. -now AxA i.e. (m x n)x(m x n) will be defined when inner dimension are same i.e. when m = n and we have already concluded that if A is square matrix then m = n hence this statement is true.