Matrices

Explanation

• Matrices may be added or subtracted only when they have the same order; combine entries in corresponding positions. Scalar multiplication multiplies every entry by the scalar.
• A product $AB$ exists when the number of columns of $A$ equals the number of rows of $B$.
• Its $(i,j)$ entry is the scalar product of row $i$ of $A$ with column $j$ of $B$, and the result has the number of rows of $A$ and columns of $B$.
• Matrix multiplication is generally not commutative, so $AB$ and $BA$ may have different orders or only one may exist.
• Dimensions should be checked before any arithmetic.

Explanation

• The zero matrix $O$ has every entry zero and is the additive identity: $A+O=A$ for matrices of the same order.
• An identity matrix $I$ is square, with ones on the leading diagonal and zeros elsewhere, and satisfies $AI=IA=A$ whenever the products are defined.
• Its order must match the matrix calculation.
• Polynomial identities in a square matrix are manipulated algebraically, but a scalar constant becomes that scalar multiplied by $I$; for example, the matrix counterpart of $x^2-5x+6$ is $A^2-5A+6I$.
• The symbols $O$ and $I$ therefore represent matrices, not the scalar numbers $0$ and $1$.

Explanation

• A linear transformation is determined by the images of the standard basis vectors, which form the columns of its matrix.
• In two dimensions, required matrices include reflections in coordinate axes or $y=\pm x$, rotations through any angle about the origin, stretches parallel to an axis and enlargements about the origin with non-zero real scale factor.
• If matrix $A$ acts first and matrix $B$ second, the combined matrix is $BA$, because $\mathbf x\mapsto A\mathbf x\mapsto BA\mathbf x$.
• In three dimensions, use a $3\times3$ matrix; transformations are confined to reflections in coordinate planes and rotations about coordinate axes.
• Basis-vector images also identify these transformations.

Explanation

• An invariant point is fixed by the transformation. For matrix $A$, its position vector satisfies $A\mathbf x=\mathbf x$, so solve $(A-I)\mathbf x=\mathbf0$.
• An invariant line is mapped onto itself as a set, although individual points on it may move.
• To test $y=mx+c$, transform a general point $(x,mx+c)$ to $(X,Y)$ and require $Y=mX+c$ for every $x$; equate coefficients and constants.
• Vertical lines must be checked separately because they cannot be written in gradient-intercept form.
• A line consisting entirely of invariant points is stronger than an invariant line, so the two descriptions must not be confused.

Explanation

• For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, $\det A=ad-bc$. A $3\times3$ determinant may be evaluated by cofactor expansion, keeping the alternating signs.
• Under a two-dimensional linear transformation, area is multiplied by $|\det A|$; in three dimensions, volume is multiplied by $|\det A|$.
• The determinant's sign records orientation: a positive determinant preserves orientation and a negative determinant reverses it.
• A zero determinant collapses dimension and gives zero area or volume scale factor.
• Geometric sizes are never negative, so use the absolute value when calculating image area or volume.

Explanation

• A square matrix is singular when its determinant is zero; it then has no inverse. A non-singular matrix has non-zero determinant and a unique inverse satisfying $AA^{-1}=A^{-1}A=I$.
• For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, $A^{-1}=\dfrac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$. A $3\times3$ inverse can be found by row-reducing $(A\mid I)$ to $(I\mid A^{-1})$, by an adjugate method or by calculator when permitted, but the process must be understood.
• Useful properties include $(AB)^{-1}=B^{-1}A^{-1}$ and $(A^{-1})^{-1}=A$.
• Matrix order matters, so inverse factors reverse order.
• Calculator output should still be checked by multiplying by the original matrix.

Explanation

• Write three linear equations in the form $A\mathbf x=\mathbf b$, where each row of $A$ contains the coefficients of $x,y,z$ in the same order and $\mathbf b$ contains the corresponding right-hand sides.
• If $A$ is non-singular, left-multiply by $A^{-1}$ to obtain $\mathbf x=A^{-1}\mathbf b$.
• The inverse may be calculated exactly or by calculator as permitted, but the matrix equation and multiplication order should be shown.
• A zero coefficient must still occupy its correct position.
• The resulting values should be checked in the original equations, since a transposed coefficient or misplaced constant can produce a plausible but incorrect matrix calculation.

Explanation

• Each linear equation in three variables represents a plane in three-dimensional space. Three independent planes meeting at one point give a unique solution.
• If all three contain the same line, they form a sheaf and every point on that line is a solution. Coincident planes can also produce infinitely many solutions.
• A prism arrangement has pairwise intersection lines but no point common to all three, so the system is inconsistent; parallel distinct planes are another no-solution case.
• Algebraically, a contradictory row after elimination signals inconsistency, while a dependent equation without contradiction leaves free variables.
• A singular coefficient matrix therefore does not by itself distinguish no solution from infinitely many solutions.