Find unknown entries in a matrix equation practice questions | General Mathematics (Year 11, VCE Units 1 & 2, Matrices, Matrix addition and subtraction)

Solving a matrix equation entry by entry

For a matrix sum or difference, compare entries in the same positions and isolate the unknown in each resulting equation. A matrix product combines a row with a column, so its equations may need to be solved together

a_{ij}+x_{ij}=c_{ij}\quad\text{or}\quad x_{ij}-b_{ij}=c_{ij}

Match positions Equal matrices have equal entries at the same row and column position \(\displaystyle A+X=C\quad\Rightarrow\quad a_{ij}+x_{ij}=c_{ij}\)
Isolate the unknown Rearrange the ordinary equation at one position without changing its operation or sign \(\displaystyle a_{ij}+x_{ij}=c_{ij}\quad\Rightarrow\quad x_{ij}=c_{ij}-a_{ij}\)
Handle subtraction If the unknown is the minuend, add the known subtracted entry to the result entry \(\displaystyle x_{ij}-b_{ij}=c_{ij}\quad\Rightarrow\quad x_{ij}=c_{ij}+b_{ij}\)
Use row and column products For a product of two-by-two matrices, form each entry from a complete row and column. The resulting equations can contain more than one unknown and may need to be solved together \(\displaystyle (XA)_{11}=x_{11}a_{11}+x_{12}a_{21}\)

Worked example

Find the unknown matrix X

X-B=C,\quad B=\begin{bmatrix}3&-2\\1&4\end{bmatrix},\quad C=\begin{bmatrix}5&6\\2&7\end{bmatrix}

\(\displaystyle X-B=C\quad\Rightarrow\quad x_{ij}-b_{ij}=c_{ij}\)Compare entries at the same row and column position
\(\displaystyle x_{11}-3=5\quad\Rightarrow\quad x_{11}=5+3=8\)Add the subtracted entry to the result entry at row 1, column 1.
\(\displaystyle x_{12}-(-2)=6\Rightarrow x_{12}=6+(-2)=4,\quad x_{21}-1=2\Rightarrow x_{21}=3\)Apply the same rearrangement to the other two positions shown
\(\displaystyle x_{22}-4=7\quad\Rightarrow\quad x_{22}=11,\quad X=\begin{bmatrix}8&4\\3&11\end{bmatrix}\)The four solved entries form the unknown matrix in their original positions

Common mistakes

Mismatching matrix positions

✕\(\displaystyle x_{12}-4=2\quad\Rightarrow\quad x_{12}=6\)✓\(\displaystyle x_{12}-(-3)=2\quad\Rightarrow\quad x_{12}=-1\)

Each unknown entry is paired with the known and result entries at the same row and column

Reversing a subtraction

✕\(\displaystyle 7-x=3\quad\Rightarrow\quad x=3-7=-4\)✓\(\displaystyle 7-x=3\quad\Rightarrow\quad x=7-3=4\)

When the unknown is subtracted from the known value, subtract the result from the known value to isolate it

Multiplying entries instead of forming a row-column product

✕\(\displaystyle (XA)_{11}=x_{11}a_{11}\)✓\(\displaystyle (XA)_{11}=x_{11}a_{11}+x_{12}a_{21}\)

For two-by-two matrices X and A, this entry uses both pairs from row 1 of X and column 1 of A, then adds the products. Its value depends on those entries

Difficulty levels

introSolve a sum equation for a 2 × 2 unknown matrix with non-negative entries

standardSolve difference or scalar-and-sum equations with signed entries

challengeSolve an unknown-first difference, scalar equation or product-equals-identity equation

examChoose one unknown entry when sign, subtraction and position errors are included as options

Questions about this topic

Why can a matrix equation be split into separate equations?+

Two matrices are equal only when their corresponding entries are equal, so each position gives its own ordinary equation

How is an unknown matrix found when a known matrix is subtracted from it?+

At each position, add the entry of the known matrix to the corresponding result entry

How is a product equation different?+

A product entry is formed from a complete row and column, so its equation contains the sum of the matching products rather than one entrywise product

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices

In this chapter

State the order of a matrix and locate its elements 4A Matrix types and order: Identify row, column, square, symmetric and zero matrices Read information from a labelled data matrix 4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matrices Add and subtract matrices 4B Matrix addition and subtraction: Find unknown entries in a matrix equation Multiply a matrix by a scalar Combine scalar multiples of matrices Decide whether a matrix product is defined and give its order Multiply matrices 4D Matrix products and powers: Evaluate powers of a square matrix Identify and construct the identity matrix 4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equation Find the determinant of a 2 × 2 matrix Find the inverse of a 2 × 2 matrix Write simultaneous equations as a matrix equation Solve simultaneous equations using the inverse matrix 4F Matrices for networks: Evaluate powers of a square matrix Represent a network in a matrix Identify the network a connection matrix describes Interpret connection sums and symmetry Count routes using powers of a connection matrix Set up a transition matrix Use a transition matrix to predict the next step 4H State matrices and recurrence: Evaluate powers of a square matrix Generate state matrices by recursion Determine the long-run state of a transition model Use matrix multiplication to answer an application question Use row and column matrices of ones to extract totals