Question

Prove each of the following laws of matrix algebra: (a) \(\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}\) (b) \(\mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C}\) (c) \(\alpha(\mathbf{A}+\mathbf{B})=\alpha \mathbf{A}+\alpha \mathbf{B}\) (d) \((\alpha+\beta) \mathbf{A}=\alpha \mathbf{A}+\beta \mathbf{A}\) (e) \(\mathbf{A}(\mathbf{B C})=(\mathbf{A B}) \mathbf{C}\) (f) \(\mathbf{A}(\mathbf{B}+\mathbf{C})=\mathbf{A B}+\mathbf{A C}\)

Short answer

Question: Prove the following laws of matrix algebra: (a) commutative law for addition; (b) associative law for addition; (c) distributive law of scalar multiplication over addition; (d) distributive law of scalar addition over matrix multiplication; (e) associative law for multiplication; (f) distributive law of matrix multiplication over addition. Answer: (a) The commutative law for matrix addition is proven through the commutative property of real numbers, showing that A + B = B + A. (b) The associative law for matrix addition is demonstrated by proving that A + (B + C) = (A + B) + C using the associative property of real numbers. (c) The distributive law of scalar multiplication over matrix addition is proven by showing that α(A + B) = αA + αB for each element. (d) The distributive law of scalar addition over matrix multiplication is demonstrated by proving that (α + β)A = αA + βA for each element. (e) The associative law of matrix multiplication is demonstrated by showing that A(BC) = (AB)C through comparison of elements. (f) The distributive law of matrix multiplication over matrix addition is proven by demonstrating that A(B + C) = AB + AC using the distributive property of real numbers.

Step-by-step solution

(a) Proving Commutative Law: A + B = B + A

Let A and B be two matrices of the same order, say m x n. We need to prove that their sum is commutative, i.e., A + B = B + A. To prove this, we will show that for every element a_ij in matrix A and every element b_ij in matrix B, the sum of the corresponding elements is equal and commutative. Since A and B are of the same order, their sum is also of order m x n. Let the sum matrix be S, where s_ij = a_ij + b_ij for 1 ≤ i ≤ m and 1 ≤ j ≤ n. Similarly, let T be the matrix representation of B + A such that t_ij = b_ij + a_ij. Now, we need to show that S = T, so we compare each element: s_ij = a_ij + b_ij = b_ij + a_ij = t_ij (by commutative property of real numbers) Since the elements of matrices S and T are equal, we can conclude that A + B = B + A.

(b) Proving Associative Law: A + (B + C) = (A + B) + C

Let A, B, and C be matrices of the same order, say m x n. To prove the associative law of matrix addition, we will show that for each element a_ij, b_ij, and c_ij, the associative property holds. Let D = B + C, and E = A + B. We need to prove that A + D = E + C. Using the definition of matrix addition, for each element: d_ij = b_ij + c_ij e_ij = a_ij + b_ij a_ij + d_ij = a_ij + (b_ij + c_ij) = (a_ij + b_ij) + c_ij = e_ij + c_ij (by the associative property of real numbers) Since the elements of matrices A + D and E + C are equal, we can conclude that A + (B + C) = (A + B) + C.

(c) Proving Distributive Law: α(A + B) = αA + αB

Let A and B be two matrices of the same order, say m x n. Let α be a scalar. We need to prove the distributive law of scalar multiplication over matrix addition. For each element, let s_ij be the (i,j)-th element of A + B, and r_ij and r'_ij be the (i,j)-th elements of αA and αB, respectively. s_ij = a_ij + b_ij α(a_ij + b_ij) = αs_ij αa_ij + αb_ij = r_ij + r'_ij The distributive property of scalar multiplication over real numbers holds for each element. We can conclude that α(A + B) = αA + αB.

(d) Proving Distributive Law: (α + β)A = αA + βA

Let A be a matrix of order m x n and α, β be scalars. We need to prove the distributive law of scalar addition over matrix multiplication. For each element a_ij: (α + β)a_ij = αa_ij + βa_ij The distributive property of real numbers holds for each element of the matrix A. We can conclude that (α + β)A = αA + βA.

(e) Proving Associative Law: A(BC) = (AB)C

Let A be an m x p matrix, B be a p x q matrix, and C be a q x n matrix. We need to prove that the product A(BC) is equal to the product (AB)C. For any matrix product, the element at position (i,k) is the dot product of the i-th row of the left matrix and the k-th column of the right matrix. Using the definition of matrix multiplication and the associative law of real numbers: [(A(BC))_ik] = Σ (a_ij (BC)_jk) = Σ (a_ij (Σ b_jr c_rk)) = Σ (Σ a_ij b_jr c_rk)= Σ (Σ (a_ij b_jr) c_rk) = [(AB)_ir] c_rk = [(AB)C]_ik Since the elements of the matrices A(BC) and (AB)C are equal, we can conclude that A(BC) = (AB)C.

(f) Proving Distributive Law: A(B + C) = AB + AC

Let A be an m x p matrix, B and C be p x n matrices. We need to prove the distributive law of matrix multiplication over matrix addition. Using the definition of matrix addition and multiplication and the distributive property of real numbers, Let D = B + C. Then, d_ij = b_ij + c_ij. [(A(B + C))_ik] = Σ a_ij d_jk = Σ a_ij (b_jk + c_jk) = Σ (a_ij b_jk + a_ij c_jk) = Σ a_ij b_jk + Σ a_ij c_jk = [(AB + AC)_ik] Since the elements of the matrices A(B + C) and AB + AC are equal, we can conclude that A(B + C) = AB + AC.

Key concepts

Matrix Addition

Matrix addition involves adding two matrices of the same dimensions by adding their corresponding elements. This operation requires that both matrices have the same number of rows and columns.

For example, if we have matrices \(\mathbf{A}\) and \(\mathbf{B}\) both of size \(m \times n\), their sum \(\mathbf{A} + \mathbf{B}\) is also an \(m \times n\) matrix. Each element in the resulting matrix is calculated by adding each corresponding element from \(\mathbf{A}\) and \(\mathbf{B}\).

This is similar to adding scalars, but it’s done element by element. Remember, matrix addition is both commutative (\(\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}\)) and associative, as long as the matrices are of compatible size.

Associative Law

The associative law for matrices states that the way matrices are grouped in addition or multiplication does not affect the result. For matrix addition, it is expressed as \(\mathbf{A} + (\mathbf{B} + \mathbf{C}) = (\mathbf{A} + \mathbf{B}) + \mathbf{C}\).

This law implies that when adding multiple matrices, you can add them in any grouping without changing the outcome. Make sure all matrices involved are of the same order.

For multiplication, this association works as \((\mathbf{A}\mathbf{B})\mathbf{C} = \mathbf{A}(\mathbf{B}\mathbf{C})\). Unlike addition, multiplication requires that the number of columns in the first matrix matches the number of rows in the second.

Distributive Law

The distributive law of matrix algebra ensures that multiplying a matrix by a scalar or another matrix distributes over addition.

Firstly, for scalar multiplication: \(\alpha(\mathbf{A} + \mathbf{B}) = \alpha \mathbf{A} + \alpha \mathbf{B}\). Here, the scalar \(\alpha\) is distributed over each individual element of the matrices \(\mathbf{A}\) and \(\mathbf{B}\), and then you sum the results.

Secondly, when you have matrix multiplication: \(\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C}\). This means that if you have a matrix multiplying a sum of two matrices, it can be distributed to multiply each matrix individually before summing.

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar (a single number). This operation scales the matrix elements uniformly.

For example, if a matrix \(\mathbf{A}\) is multiplied by a scalar \(\alpha\), then every element \(a_{ij}\) of \(\mathbf{A}\) is multiplied by \(\alpha\). The resulting matrix is the same size as \(\mathbf{A}\), but each element has been scaled by \(\alpha\).

Scalar multiplication is often used in combination with addition and subtraction in matrix equations to help solve systems of linear equations. It’s a straightforward operation, but remember that it affects every single entry of the matrix, making it a powerful tool in matrix algebra.