MATRICES LEC3

Adjoint of a Matrix: The Short Trick Method

• Introduction to the Adjoint Calculation:

  • The adjoint of a matrix is a critical component for finding the inverse, which is often tested in examinations.

  • While the traditional method involving cofactors ($a_{11}, a_{12}, a_{13}, \dots$) followed by a transpose is lengthy and prone to calculation errors, a shortcut method exists that provides the answer within 2 to 3 minutes.

  • Calculators often cannot help with these specific structural steps in symbolic or complex matrix problems, making the shortcut essential.

• Step-by-Step Short Trick for a $3 \times 3$ Adjoint:

  • Step 1: Write the Original Matrix: Let the matrix $A$ be:       $\begin{pmatrix} 1 & 3 & 1 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{pmatrix}$

  • Step 2: Expand Rows and Columns: Write down the matrix and then copy the first two rows below it. Then, take that entire expanded block and copy the first two columns to the right.

  • Step 3: Eliminate Borders: Strike out the first (top) row and the first (extreme left) column. You are left with a $4 \times 4$ grid of numbers.

  • Step 4: Cross-Multiply in Pairs: Perform $2 \times 2$ determinant-style cross-multiplication on adjacent elements.

    • For the first column of the adjoint: $(4 \times 4) - (3 \times 3) = 16 - 9 = 7$. Next, $(3 \times 1) - (4 \times 1) = 3 - 4 = -1$. Then, $(1 \times 3) - (3 \times 1) = 3 - 3 = 0$. (Note: Calculate vertically but write the results horizontally, or vice versa, to account for the transpose).

  • Step 5: Final Result: Following this process for the matrix provided in the example yields values that, once transposed, form the Adjoint. For example: $\begin{pmatrix} 7 & -9 & 5 \\ -1 & 3 & -2 \\ -1 & 0 & 1 \end{pmatrix}$ (values may vary based on specific transcript arithmetic).

• Example 2 of Adjoint Shortcut:

  • Matrix elements: $1, 2, 3$; $0, 1, 4$; $5, 2, 0$.

  • Method: Repeat first two rows and first two columns. Eliminate the top row and left column.

  • Calculations:

    • $(1 \times 0) - (2 \times 4) = -8$

    • $(4 \times 5) - (0 \times 0) = 20$

    • $(0 \times 2) - (1 \times 5) = -5$

  • Continuing this yields the adjoint matrix elements through rapid $2 \times 2$ subtraction.

Identity and Inverse Properties in Matrices

• Identity Property Definition:

  • An identity property exists when an element $a$ is multiplied by an identity element $e$ such that the result is the original element $a$.

  • In numbers: $2 \times 1 = 2$. Here, $1$ is the multiplicative identity.

  • In matrices: If Matrix $A$ is multiplied by the Unit Matrix ($I$), the result is Matrix $A$. This is the Multiplicative Identity Property ($A \times I = A$).

• Additive Identity:

  • For numbers, the additive identity is $0$ (e.g., $2 + 0 = 2$).

  • In matrices, adding the Null Matrix ($O$) to Matrix $A$ results in Matrix $A$ ($A + O = A$).

• Inverse Properties:

  • A number multiplied by its inverse yields the identity element (e.g., $2 \times \frac{1}{2} = 1$).

  • Multiplicative Inverse: For a matrix $A$, there exists an inverse $A^{-1}$ such that $A \times A^{-1} = I$.

  • Additive Inverse: Adding a matrix to its negative ($-A$) results in the Null Matrix ($A + (-A) = O$).

• Conditions for Inversion:

  • A matrix must be non-singular for an inverse to exist.

  • If the determinant is zero ($det(A) = 0$), the matrix is singular, and the multiplicative inverse does not exist.

Systems of Linear Equations

• Matrix Representation of Equations:

  • Equations like $2x + 3y + 4z = 1$, $3x + y + 6z = 4$, and $x + y + z = 11$ can be converted into the matrix form $Ax = B$.

  • Coefficient Matrix ($A$): Capture the coefficients of variables. Always follow alphabetical order ($x, y, z$).

    • Example: Row 1 = $(2, 3, 4)$, Row 2 = $(3, 1, 6)$, Row 3 = $(1, 1, 1)$.

  • Variable Matrix ($X$): A column matrix of variables $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$.

  • Constant Matrix ($B$): A column matrix of values on the right side of the equals sign $\begin{pmatrix} 1 \\ 4 \\ 11 \end{pmatrix}$.

• Classification of Systems:

  • Homogeneous System: If the constant matrix $B$ is a Null Matrix ($B = 0$).

  • Non-Homogeneous System: If the constant matrix $B$ contains at least one non-zero element ($B \neq 0$).

General Laws and Properties of Matrix Operations

• Commutative and Associative Laws:

  • Addition: Matrix addition is commutative ($A + B = B + A$).

  • Multiplication: Matrix multiplication is NOT commutative ($AB \neq BA$).

  • Associative Law: Matrix multiplication is associative ($A(BC) = (AB)C$).

• Transpose and Inverse Laws:

  • Transpose of a Transpose: $(A^T)^T = A$.

  • Inverse of an Inverse: $(A^{-1})^{-1} = A$.

  • Sum Transpose: $(A+B)^T = A^T + B^T$.

  • Product Transpose (Reversal Law): $(AB)^T = B^T A^T$. For three matrices, $(ABC)^T = C^T B^T A^T$.

  • Scalar Multiplication and Transpose: If $k$ is a constant, $(kA)^T = k(A^T)$.

  • Product Inverse: $(AB)^{-1} = B^{-1} A^{-1}$.

  • Distributive Property: $A(B+C) = AB + AC$.

Determinant Shortcuts and Calculations

• Triangular Matrix Shortcut:

  • If a matrix is Upper Triangular, Lower Triangular, or Diagonal (elements only on the main diagonal and zeros above/below), the determinant is simply the product of the diagonal elements.

  • Example: Determinant of $\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} = abc$.

• Zero Determinant Trick:

  • If any two rows or columns are identical, the determinant is $0$.

  • Arithmetic Check: If subtracting Row 1 from Row 2 ($R_2 - R_1$) yields the same result as subtracting Row 2 from Row 3 ($R_3 - R_2$), the determinant is often $0$.

• Common Factors:

  • You can take a common factor out of an individual row or column in a determinant calculation. This simplifies large numbers.

  • Example: In $\begin{pmatrix} y^2 & y & 1 \\ 8 & 4 & 10 \\ 9 & 3 & 6 \end{pmatrix}$, you can take $2$ out of the second row and $3$ out of the third row to simplify the math.

• Matrix Scaling and Determinants:

  • If a matrix $A$ of order $n \times n$ is multiplied by a scalar $k$, the determinant of the new matrix is:       $det(kA) = k^n \times det(A)$

  • Problem Example: If matrix $A$ is of order $3$ and $det(A) = 2$, find $det(2A)$.

    • Method: Since order $n=3$ and scalar $k=2$, the answer is $2^3 \times 2 = 8 \times 2 = 16$.

• Minor Elements and Order Reduction:

  • Deleting the $i^{th}$ row and $j^{th}$ column of an $n \times n$ matrix results in a Minor Element with a reduced order of $(n-1) \times (n-1)$.

Questions & Discussion

• Q: Why multiply every element by 2 in the determinant problem?

  • A: When we say $2A$, we are multiplying the entire matrix by the scalar $2$. In a $3 \times 3$ matrix, this means all 9 elements are doubled. When calculating the determinant, a factor of 2 can be pulled out from each of the 3 rows, resulting in $2 \times 2 \times 2 = 8$ multiplied by the original determinant value.

• Q: Can we solve for variables like 'y' directly from a determinant?

  • A: Yes. By expanding the determinant, you create an equation. If the highest power of 'y' is 2, it forms a quadratic equation, meaning there will be two possible values for 'y'. You can verify these values by plugging the options from a multiple-choice question back into the simplified matrix.

• Q: When should we apply properties versus direct expansion?

  • A: If a matrix has 0s or 1s, or shows clear patterns (arithmetic progressions in rows), properties are faster. In complex engineering exams (like ECAT or Aptitude tests), properties save significant time. If no pattern is obvious, direct expansion is the fallback.