MATH1061 Chap.7 Matrices
Matrices
Matrices represent linear transformations and organise systems of linear equations. This chapter connects each algebraic operation to that role: a matrix-vector product combines columns to produce an output, matrix multiplication composes transformations and row reduction preserves a system's solution set while exposing pivots. Determinants and inverses are used as structural tests, not detached formulas.
Order and dimensions are checked before multiplication because composition is generally not commutative. A row-by-column product computes one output entry at a time, while the column view explains the entire output as a linear combination. Elementary row operations preserve the solution set but change the displayed equations into a form where pivots and free variables are visible.
For square matrices, a nonzero determinant connects unique solvability, independent columns and the existence of an inverse. Every calculation ends with a structural check: expected dimensions, the identity product or substitution into the original system.
What this chapter covers
- 01
Interpret matrices as linear maps
- 02
Check dimensions before multiplication
- 03
Compute matrix-vector and matrix-matrix products
- 04
Solve systems with row operations
- 05
Use determinants to test invertibility
- 06
Verify inverse and solution calculations
Worked example · free
Solve a two-equation linear system
- 1Multiply the first equation by 3 to obtain 3x + 6y = 15.
- 1Subtract the second equation to get 7y = 11.
- 1Thus y = 11/7.
- 1Substitute into the first equation to obtain x = 13/7.
- 1Substitution gives 5 and 4 on the two left sides, confirming the solution.
Key terms
- Linear map
- A transformation preserving vector addition and scalar multiplication.
- Matrix product
- The row-by-column operation representing composition of compatible linear maps.
- Pivot
- A leading entry used to eliminate other entries in its column.
- Determinant
- A scalar recording signed scale change and whether a square matrix is invertible.
- Row operation
- A reversible equation operation applied to every entry across a matrix row.
- Free variable
- A variable associated with a non-pivot column in a consistent reduced system.
- Identity matrix
- A square matrix that leaves compatible vectors and matrices unchanged under multiplication.
- Matrix transpose
- The matrix formed by exchanging rows and columns.
Matrices FAQ
Why does matrix multiplication order matter?
A product represents composition of transformations, and changing the order changes which transformation acts first. The dimensions can also make one order valid while the reverse order is impossible. Write the input and output dimensions above each factor before multiplying; this exposes compatibility and helps preserve the intended composition.
What does a zero determinant tell me?
For a square matrix, a zero determinant means the transformation collapses at least one direction and is not invertible. The corresponding columns are linearly dependent, and a system with that coefficient matrix cannot have a unique solution. The determinant diagnoses structure, while row reduction reveals whether the system has none or infinitely many solutions.
How should I check a solved linear system?
Substitute the proposed values into every original equation. If an inverse was used, multiply the matrix by the claimed inverse and look for the identity. If row reduction was used, make sure each operation was applied across the entire row, including the augmented entry. These checks catch most arithmetic errors without repeating the full method.
What do free variables mean in a reduced system?
A free variable corresponds to a non-pivot column and can vary while the equations remain satisfied. If the system is consistent, express pivot variables in terms of the free variables to describe infinitely many solutions. A contradictory row instead signals no solution. Counting equations alone cannot distinguish those outcomes; the reduced rows reveal the structure.
Exam move
Begin every product by writing dimensions and the expected output shape. For matrix-vector multiplication, describe each output entry as a row dot product and also view the result as a combination of columns. Alternate multiplication drills with transformation questions so order remains meaningful rather than mechanical.
Practise elimination with the row operation written beside every step so errors remain traceable, and apply each operation to the augmented entry as well as the coefficient block. After reaching echelon form, identify pivots, free variables and any contradictory row before solving backward.
Link determinants to geometry and uniqueness, then verify numerical answers by multiplication or substitution rather than trusting a long elimination chain. For a proposed inverse, test both the expected dimensions and the identity product. When matrices are composed, read the rightmost action first and use a sample vector to confirm the order. These checks expose structural mistakes earlier than recomputing every entry.
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