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Matrix Operations

Matrix operations are the algebraic procedures performed on matrices. These include addition, subtraction, scalar multiplication, and matrix multiplication.

Addition and Subtraction of Matrices

Two matrices can be added or subtracted only if they have the same order (i.e., the same number of rows and the same number of columns). The operation is performed by adding or subtracting the corresponding elements.

If \(A = [a_{ij}]\) and \(B = [b_{ij}]\) are two matrices of order \(m \times n\), then their sum \(A+B\) is a matrix \(C = [c_{ij}]\) of the same order, where \(c_{ij} = a_{ij} + b_{ij}\) for all \(i\) and \(j\).

  • Example: Let \(A = \begin{bmatrix} 8 & 3 \\ 4 & 5 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & -2 \\ -4 & 6 \end{bmatrix}\).

    • Addition:

      \[ A+B = \begin{bmatrix} 8+1 & 3+(-2) \\ 4+(-4) & 5+6 \end{bmatrix} = \begin{bmatrix} 9 & 1 \\ 0 & 11 \end{bmatrix} \]

    • Subtraction:

      \[ A-B = \begin{bmatrix} 8-1 & 3-(-2) \\ 4-(-4) & 5-6 \end{bmatrix} = \begin{bmatrix} 7 & 5 \\ 8 & -1 \end{bmatrix} \]

Properties of Matrix Addition

  • Commutative Law: \(A + B = B + A\)
  • Associative Law: \((A + B) + C = A + (B + C)\)
  • Existence of Additive Identity: The zero matrix \(O\) is the additive identity. \(A + O = O + A = A\).
  • Existence of Additive Inverse: For any matrix \(A\), there is a matrix \(-A\) such that \(A + (-A) = (-A) + A = O\).

Scalar Multiplication

Multiplying a matrix by a scalar (a single number) involves multiplying every element of the matrix by that scalar.

If \(A = [a_{ij}]\) is a matrix of order \(m \times n\) and \(k\) is a scalar, then \(kA\) is another matrix of the same order, obtained by multiplying each element of \(A\) by \(k\). So, \(kA = [ka_{ij}]\).

  • Example: Let \(A = \begin{bmatrix} 3 & 1 \\ -2 & 0 \\ 5 & 4 \end{bmatrix}\) and \(k = -2\).

    \[ -2A = -2 \begin{bmatrix} 3 & 1 \\ -2 & 0 \\ 5 & 4 \end{bmatrix} = \begin{bmatrix} -2(3) & -2(1) \\ -2(-2) & -2(0) \\ -2(5) & -2(4) \end{bmatrix} = \begin{bmatrix} -6 & -2 \\ 4 & 0 \\ -10 & -8 \end{bmatrix} \]

Properties of Scalar Multiplication

If \(k\) and \(l\) are scalars and \(A\) and \(B\) are matrices of the same order:

  • \(k(A + B) = kA + kB\)
  • \((k + l)A = kA + lA\)

Matrix Multiplication

The product of two matrices \(A\) and \(B\), denoted \(AB\), is defined only if the number of columns in \(A\) is equal to the number of rows in \(B\) L.

If \(A\) is an \(m \times n\) matrix and \(B\) is an \(n \times p\) matrix, then their product \(AB\) is an \(m \times p\) matrix.

[insert image on matrix multiplication rule columns equal rows here]

To find the element in the \(i\)-th row and \(j\)-th column of the product matrix \(AB\), you take the dot product of the \(i\)-th row of \(A\) with the \(j\)-th column of \(B\).

  • Example: Let \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\) (\(2 \times 3\)) and \(B = \begin{bmatrix} 7 & 8 \\ 9 & 1 \\ 2 & 3 \end{bmatrix}\) (\(3 \times 2\)). The product \(AB\) will be a \(2 \times 2\) matrix.

    \[ AB = \begin{bmatrix} (1\cdot7 + 2\cdot9 + 3\cdot2) & (1\cdot8 + 2\cdot1 + 3\cdot3) \\ (4\cdot7 + 5\cdot9 + 6\cdot2) & (4\cdot8 + 5\cdot1 + 6\cdot3) \end{bmatrix} \]
    \[ AB = \begin{bmatrix} (7 + 18 + 6) & (8 + 2 + 9) \\ (28 + 45 + 12) & (32 + 5 + 18) \end{bmatrix} = \begin{bmatrix} 31 & 19 \\ 85 & 55 \end{bmatrix} \]

Properties of Matrix Multiplication

  • Not Commutative (Generally): In most cases, \(AB \neq BA\). In the example above, \(BA\) would be a \(3 \times 3\) matrix, which is clearly not equal to the \(2 \times 2\) matrix \(AB\).
  • Associative Law: \((AB)C = A(BC)\), provided the products are defined.
  • Distributive Law:
    • \(A(B+C) = AB + AC\)
    • \((A+B)C = AC + BC\)
  • Existence of Multiplicative Identity: For every square matrix \(A\), there is an identity matrix \(I\) of the same order such that \(AI = IA = A\).

Elementary Row and Column Operations

These are fundamental operations used to manipulate matrices, often for solving systems of linear equations or finding a matrix inverse. There are three types of elementary operations.

1. Interchange (Swapping)

Any two rows (or columns) of a matrix can be interchanged.

  • Notation: \(R_i \leftrightarrow R_j\) denotes the interchange of the \(i\)-th and \(j\)-th rows.

2. Multiplication (Scaling)

The elements of any row (or column) can be multiplied by a non-zero scalar.

  • Notation: \(R_i \to kR_i\) denotes multiplying the \(i\)-th row by the scalar \(k\).

3. Addition (Combination)

You can add a scalar multiple of one row (or column) to another row (or column).

  • Notation: \(R_i \to R_i + kR_j\) denotes adding \(k\) times the \(j\)-th row to the \(i\)-th row.

  • Example of Row Operations:

    Let \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\).

    • Applying \(R_2 \leftrightarrow R_3\):
    \[ \begin{bmatrix} 1 & 2 & 3 \\ 7 & 8 & 9 \\ 4 & 5 & 6 \end{bmatrix} \]

    • Applying \(R_2 \to R_2 - 4R_1\) to the original matrix \(A\):

      \[ R_2: \begin{bmatrix} 4 & 5 & 6 \end{bmatrix} - 4 \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 4-4 & 5-8 & 6-12 \end{bmatrix} = \begin{bmatrix} 0 & -3 & -6 \end{bmatrix} \]

      Resulting matrix:

      \[ \begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 7 & 8 & 9 \end{bmatrix} \]

Vector Operations

Vectors can be represented as single-row (row vectors) or single-column (column vectors) matrices. Therefore, vector addition, subtraction, and scalar multiplication are simply special cases of the matrix operations defined above.

Problem-set

Section A: Matrix Addition, Subtraction & Scalar Multiplication

  1. Given matrices \(A = \begin{pmatrix} 2 & 4 \\ 3 & 1 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 0 \\ -1 & 5 \end{pmatrix}\), find \(A + B\).
  2. Given matrices \(P = \begin{pmatrix} 7 & -2 \\ 4 & 5 \end{pmatrix}\) and \(Q = \begin{pmatrix} 3 & 3 \\ 1 & 0 \end{pmatrix}\), find \(P - Q\).
  3. If \(C = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & -1 \end{pmatrix}\), find \(3C\).
  4. Using matrices \(A = \begin{pmatrix} 2 & 4 \\ 3 & 1 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 0 \\ -1 & 5 \end{pmatrix}\), compute \(2A - B\).
  5. Verify the commutative property of addition (\(A+B = B+A\)) for \(A = \begin{pmatrix} 1 & 5 \\ 2 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} 0 & -2 \\ 3 & -1 \end{pmatrix}\).
  6. Find matrix \(X\) such that \(2X + A = B\), where \(A = \begin{pmatrix} 4 & 0 \\ -2 & 2 \end{pmatrix}\) and \(B = \begin{pmatrix} 2 & -2 \\ 4 & 6 \end{pmatrix}\).
  7. If \(k=4\), \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\), and \(B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), show that \(k(A+B) = kA + kB\).
  8. Given \(A = \begin{pmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{pmatrix}\) and \(B = \begin{pmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{pmatrix}\), find the matrix \(X\) such that \(2A + 3X = 5B\).

Section B: Matrix Multiplication

  1. Let \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}\). Calculate \(AB\).
  2. Using the matrices from Q9, calculate \(BA\) and show that \(AB \neq BA\).
  3. Let \(C = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \end{pmatrix}\) and \(D = \begin{pmatrix} 3 & 1 \\ 2 & 1 \\ 1 & 0 \end{pmatrix}\). Compute \(CD\).
  4. Is the product \(DC\) defined for the matrices in Q11? If yes, compute it.
  5. Given \(A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\), \(B = \begin{pmatrix} 2 & 0 \\ 3 & 4 \end{pmatrix}\), and \(C = \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}\), verify the associative property \((AB)C = A(BC)\).
  6. Using the matrices from Q13, verify the distributive property \(A(B+C) = AB + AC\).
  7. If \(A = \begin{pmatrix} 3 & -2 \\ 4 & -2 \end{pmatrix}\), find a scalar \(k\) such that \(A^2 = kA - 2I\), where \(I\) is the \(2 \times 2\) identity matrix.
  8. If \(A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\), find \(A^2\) and \(A^3\).
  9. Find two non-zero \(2 \times 2\) matrices \(A\) and \(B\) such that their product \(AB\) is the zero matrix \(O\).
  10. If \(f(x) = x^2 - 5x + 6\), find \(f(A)\) where \(A = \begin{pmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{pmatrix}\).

Section C: Elementary Row and Column Operations

For questions 19-24, use the matrix \(M = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\).

  1. Apply the row operation \(R_1 \leftrightarrow R_3\) to matrix \(M\).
  2. Apply the column operation \(C_2 \leftrightarrow C_3\) to matrix \(M\).
  3. Apply the row operation \(R_2 \to 3R_2\) to matrix \(M\).
  4. Apply the column operation \(C_1 \to \frac{1}{2}C_1\) to matrix \(M\).
  5. Apply the row operation \(R_2 \to R_2 - 4R_1\) to matrix \(M\).
  6. Apply the column operation \(C_3 \to C_3 - 2C_2\) to matrix \(M\).
  7. On matrix \(A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 1 & -1 \\ 3 & 2 & 0 \end{pmatrix}\), perform the operations \(R_2 \to R_2 - 2R_1\) and then \(R_3 \to R_3 - 3R_1\).
  8. Transform the matrix \(B = \begin{pmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\ 3 & 7 & 11 \end{pmatrix}\) by making the element in the first row, first column a '1' using a scaling operation.
  9. Using the result from Q26, perform row operations to create zeros below the leading '1' in the first column.
  10. A matrix \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) is transformed into \(B = \begin{pmatrix} 1 & 2 \\ 0 & -2 \end{pmatrix}\). What elementary row operation was applied?
  11. What single elementary row operation will transform the matrix \(C = \begin{pmatrix} 1 & 5 & 2 \\ 0 & 1 & 7 \\ 0 & 0 & 3 \end{pmatrix}\) into an upper triangular matrix with only 1s on the diagonal?
  12. Given \(A = \begin{pmatrix} 1 & 0 \\ -1 & 3 \end{pmatrix}\), find the matrix obtained by applying \(R_2 \to R_2 + R_1\) followed by \(C_1 \to C_1 - C_2\).